Analysis on manifolds · 1.5. Linear functional analysis 36 1.6. Differentiable calculus on Banach spaces 41 1.7. Calculus of variations 48 1.8. Implicit function theorem and inverse

Analysis on manifolds

Yi Li

SCHOOL OF MATHEMATICS AND SHING-TUNG YAU CENTER, SOUTHEASTUNIVERSITY, NANJING, CHINA

E-mail address: [email protected]; [email protected]; [email protected]

Contents

Chapter 1. Basic analysis 51.1. Set theory 51.2. Algebraic structures 71.3. Topology 101.4. Measures and integrations 211.5. Linear functional analysis 361.6. Differentiable calculus on Banach spaces 411.7. Calculus of variations 481.8. Implicit function theorem and inverse function theorem 501.9. Differentiable equations 541.10. Problems and references 57

Chapter 2. Differentiable manifolds 612.1. Differentiable manifolds 612.2. Vector fields and tensor fields 652.3. Lie groups 782.4. Exterior differential forms 782.5. Integration 782.6. Exercises and problems 78

3

CHAPTER 1

Basic analysis

1.1. Set theory

Given a set X. Let A be a subset of X. The characteristic function of A is

χA(x) :=

1, x ∈ A,0, x /∈ A.

A partition of X is a family Xii∈I of subsets such that

Xi 6= ∅, Xi ∩ Xj = ∅ (i 6= j),⋃i∈I

Xi = X.

Here I is the indexed set which may be uncountable.

1.1.1. Categories. A category C is a class consisting of

Ob(C) = objects such like X, Y, · · · , ,Mor(C) = sets HomC(X, Y) : X, Y ∈ Ob(C) ,

where elements of HomC(X, Y) are called morphisms, together with composi-tions:

HomC(X, Y)×HomC(Y, Z) −→ HomC(X, Z), ( f , g) 7−→ g f ,

satisfying the following conditions:(i) h (g f ) = (h g) f ,

(ii) for each X ∈ Ob(C) there is a unique element 1X ∈ HomC(X, X) suchthat

f 1X = f , 1X g = g,for any f ∈ HomC(X, Y) and any g ∈ HomC(Y, X).

The simplest example is Set consisting of sets and set mappings.

A covariant functor C : A→ B consists of mappings

C : Ob(A) −→ Ob(B), X 7−→ C(X)

and

C : Mor(A) −→ Mor(B), f ∈ HomA(X, Y) 7−→ C( f ) ∈ HomB(C(X), C(Y)),

such thatC(g f ) = C(g) C( f ), C(1X) = 1C(X).

A contracovariant functor C : A→ B consists of mappings

C : Ob(A) −→ Ob(B), X 7−→ C(X)

and

C : Mor(A) −→ Mor(B), f ∈ HomA(X, Y) 7−→ C( f ) ∈ HomB(C(Y), C(X)),

5

6 1. BASIC ANALYSIS

such thatC(g f ) = C( f ) C(g), C(1X) = 1C(X).

1.1.2. Relations. A relation between sets X and Y is a subset R of X×Y. Write

(x, y) ∈ R ⇐⇒ xRy.

A relation R ⊂ X× X is an equivalence relation in X if

(i) (reflexive) ∀ x ∈ X =⇒ (x, x) ∈ R;(ii) (symmetric) ∀ x, y ∈ X and (x, y) ∈ R =⇒ (y, x) ∈ R;

(iii) (transitive) ∀ x, y, z ∈ X and (x, y) ∈ R, (y, z) ∈ R =⇒ (x, z) ∈ R.

If R is an equivalence relation, we write x ∼ y instead of xRy.

(1) The equivalence class of x ∈ X is

[x] := y ∈ X : y ∼ x.

Then X is the disjoint union of all [x] and denote

X/ ∼:= [x] : x ∈ X.

(2) Define x ∈ y in R by x− y = 2πn for some n ∈ Z. Then R/ ∼∼= S1 theunit circle on R2.

1.1.3. Orderings. A relation R ⊂ X× X is a partial ordering in X if

(i) (reflexive) ∀ x ∈ X =⇒ (x, x) ∈ R;(ii) (anti-symmetric) (x, y) ∈ R and (y, x) ∈ R =⇒ x = y;

(iii) (transitive) (x, y) ∈ R and (y, z) ∈ R =⇒ (x, z) ∈ R.

If R is a partial ordering, we write x ≤ y instead of xRy.

(1) A partially ordered set is a pair (P,≤), where ≤ is a partial ordering onP. Let a, b, c ∈ P.

(1.1) c is an upper bound for a and b if a ≤ c and b ≤ c.(1.2) c is the least upper bound or supremum of a and b, if c is an upper

bound for a and b, and satisfies c ≤ d for any upper bound d for aand b. Write

c = sup(a, b) = a ∨ b.

(1.3) Similarly, one can define a lower bound for a and b and the greatestlower bound or infimum c of a and b. Write

c = inf(a, b) = a ∧ b.

(2) A partially ordered set (P,≤) is directed if any pair of elements of P hasan upper bound.

(3) A partially ordered set (P,≤) is a lattice if for any a, b ∈ P, both sup(a, b)and inf(a, b) exist.

(4) Let (P,≤) is a partially ordered set, and m, p ∈ P. We say m is maximalif m ≤ p⇒ m = p.

(5) A partially ordered set (P,≤) is linearly ordered or totally ordered if forany a, b ∈ P, either a ≤ b or b ≤ a.

1.2. ALGEBRAIC STRUCTURES 7

Lemma 1.1.1. (Zorn’s Lemma) Let (P,≤) be a nonempty partially ordered set such thateach linearly ordered subset of P has an upper bound in P. Then for every x ∈ P there is amaximal element y ∈ P such that x ≤ y.

Exercise 1.1.2. (1) P = Z+ = N \ 0, and m ≤ n⇔m|n. Then (P,≤) is a partiallyordered set, and

sup(m, n) = [m, n], inf(m, n) = (m, n).(2) P = R and x ≤ y in the usual sense. Then (P,≤) is a partially ordered set,

andsup(x, y) = maxx, y, inf(x, y) = minx, y.

(3) P = 2U , where U is a set, and A ≤ B⇔ A ⊆ B. Then (P,≤) is a partiallyordered set and

sup(A, B) = A ∪ B, inf(A, B) = A ∩ B.(4) Given a nonempty set X, define P the set of all real functions on X, and

f ≤ g⇔ f (x) ≤ g(x) for all x ∈ X. Then (P,≤) is a partially ordered set.

1.2. Algebraic structures

Let A be a subset of a set X. An internal operation on X is a mapping X×X →X, while an external operation on X by A is a mapping A× X → X.

1.2.1. Groups. A groups is a set X together with an internal operations (calledmultiplication)

X× X −→ X, (x, y) 7−→ xy,such that

(i) (associative) ∀ x, y, z ∈ X =⇒ (xy)z = x(yz);(ii) (identity) ∃ (hence ∃!) e ∈ X such that xe = ex = x for all x ∈ X;

(iii) (inverse) ∀ x ∈ X ∃! x−1 ∈ X such that xx−1 = e = x−1x.The group X is Abelian if xy = yx for all x, y ∈ X.

(1) Define a category Group as follows: objects are groups, and morphismsare group homomorphisms. That is, f ∈ HomGroup(G1, G2) if and onlyif f (x1y1) = f (x1) f (x2) for any x1, y1 ∈ G1.

(2) Let X be a group.(2.1) The center of X is the set x ∈ X : xy = yx, ∀ y ∈ X. It clearly

contains e.(2.2) A subset A ⊆ X is said to be a subgroup of X if xy−1 ∈ A whenever

x, y ∈ A. In this case we write A < X.(2.3) Suppose A < X and choose x ∈ X. Define the left coset and right

coset by

xA := xa : a ∈ A, Ax := ax : a ∈ A,respectively.

(2.4) A < X is said to be normal if xax−1 ∈ A for all a ∈ A and all x ∈ X.In this case we have Ax = xA and write A / X.

8 1. BASIC ANALYSIS

(2.5) For A / X define the quotient group by

X/A := xA : x ∈ Awith multiplication (xA)(yA) := (xy)A.

Exercise 1.2.1. (1) (Max(n, R),+) is an Abelian group, where Max(n, R) is the setof all n× n real matrices.

(2) (GL(n, R),+) is a non-Abelian group, where GL(n, R) is the subset ofMax(n, R) with nonzero determinants.

(3) T := R/Z ∼= S1.(4) Show that Group is a category.

1.2.2. Rings. A ring is a set X together with multiplication (x, y) 7→ xy, andaddition (x, y) 7→ x + y, such that

(i) (X,+) is Abelian (so has the identity 0);(ii) (associative and distributive):

(xy)z = x(yz), x(y + z) = xy + xz, (y + z)x = yx + zx.

A ring with an identity e (i.e., ex = xe = x for all x ∈ X) is called a ring withidentity.

(1) A ring is Abelian under multiplication is said to be a commutative ring.(2) Let X be a ring with identity. x ∈ X is said to be invertible if ∃ y ∈ X

such that xy = e = yx (show that such a y is unique!).(3) A ring with identity is called a field if ∀ 0 6= X is invertible.(4) Usually we take K = R or C.

Let X be a ring.(1) A left ideal I of X is a subring of X such that ∀ x ∈ X and ∀ i ∈ I ⇒

xi ∈ I.(2) Similarly one can define a right ideal of X. An ideal of X is both left and

right.(3) Let I be an ideal of X. Define an equivalence relation by

x ∼ y ⇐⇒ x− y ∈ I.

Then the quotient

X/I :=[x] := x + i|i ∈ I

∣∣x ∈ X

is a ring, called a quotient ring of X:

[x][y] := [xy], [x] + [y] := [x + y].

1.2.3. Modules. A module X over R (ring) is an Abelian group X togetherwith an external operation (scalar multiplication):

R× X −→ X, (α, x) 7−→ αx,

such that(i) α(x + y) = αx + αy, ∀ x, y ∈ X and ∀ α ∈ R,

(ii) (α + β)x = αx + βx, ∀ x ∈ X and ∀ α, β ∈ R,(iii) (αβ)x = α(βx), ∀ x ∈ X and ∀ α, β ∈ R,

1.2. ALGEBRAIC STRUCTURES 9

(iv) ex = x, ∀ x ∈ R (if R is a ring with identity).

1.2.4. Algebras. An algebra A is a module over a ring R with identity, to-gether with an internal associative operation (called multiplication) such that

(i) A is a ring,(ii) α(xy) = (αx)y = x(αy), ∀ x, y ∈ A and ∀ α ∈ R.

1.2.5. Linear spaces. A linear space or vector space is a module X (over ringR) for which the ring of operators is a field (namely, R is a field). Usually andalways, we take R = K. Elements of X are called vectors. A subset L of X is calleda vector subspace if L is a module over the field K.

(1) Let X be a vector space, and L, M are vector subspaces. We say X is thedirect sum of L and M,

X = L⊕M,

if ∀ z ∈ X ∃! x ∈ L and y ∈ M such that z = x + y.(2) Let X be a vector space.

(2.1) A ⊂ X is called linearly independent if ∀ xi1≤i≤n ⊂ A with∑1≤i≤n λixi = 0, then λi = 0 for each 1 ≤ i ≤ n.

(2.2) A Hamel basis of X is a maximal linearly independent subset of X(the existence follows from Lemma 1.1.1).

(2.3) Two Hamel bases have the same cardinality, so that we can definethe dimension of X. That is, dim(X) := #Hamel basis.

(2.4) The codimension of a vector subspace L ⊂ X is codim(L) := dim(X \L).

(2.5) A subset A of X is convex if

x, y ∈ A, 0 ≤ λ ≤ 1 =⇒ λx + (1− λ)y ∈ A.

(2.6) An affine subspace of affine hyperplane of X is a set

x ∈ X : x = y + x0, y ∈ L,where x0 is a given vector of X and L is a vector subspace of X.

Let X, Y be vector spaces over K.(1) A mapping f : X → Y is linear if

f (αx + βy) = α f (x) + β f (y)

for all x, y ∈ X and all α, β ∈ K. Define

Ker( f ) := x ∈ X : f (x) = 0.

Theorem 1.2.2. A linear mapping f is injective if and only if ker( f ) = 0.

Theorem 1.2.3. The inverse of a bijective linear mapping is also linear.

(2) Define

(1.2.1) L(X, Y) := linear mappings from X to Y.

10 1. BASIC ANALYSIS

(3) Set

(1.2.2) X∗ := L(X, K)

the algebraic dual of X. Elements of X∗ are called linear forms or linearfunctionals.

(4) A sesquilinear mapping is a mapping

X× X −→ K, (x, y) 7−→ (x|y),satisfying

(x|y) = (y|x), (αx + βy|z) = α(x|z) + β(y|z).(5) A sesquilinear mapping is nondegenerate if

f : X −→ X∗, x 7−→ (x|·)is bijective. If dim(X) is finite, then

(1.2.3) nondegenerate ⇐⇒((y|x) = 0, ∀ x ∈ X ⇒ y = 0

).

(6) A sesquilinear mapping is positive if (x|x) ≥ 0, ∀ x ∈ X. It is strictlypositive if it is positive and ((x|x) = 0⇒ x = 0).

(7) A pre-Hilbert space is a pair (X, (·|·)), where X is a vector space and (·|·)is a strictly positive sesquilinear mapping.

Exercise 1.2.4. (1) Show that L(X, Y) is a vector space over K.(2) On X = R2 define

(x|y) := x1y1, x = (x1, x2), y = (y1, y2) ∈ R2.

Show that (·|·) i a degenerate sesquilinear mapping.(3) On X = C(U) (the set of all continuous functions on U), where U is a closed

and bounded interval of R, define

(y|x) :=∫

Uy(t)x(t)dt.

Show that dim(X) = ∞, (·|·) is a degenerate sesquilinear mapping, and (1.2.3) isnot true in this case.

1.3. Topology

1.3.1. Topology. A system T of subsets of a set X defines a topology on X if(i) ∅, X ∈ T ,

(ii) ∀ Ui ∈ T , i ∈ I =⇒ ∪i∈IUi ∈ T ,(iii) ∀ U1, · · · , Uk ∈ T =⇒ ∩1≤i≤kUi ∈ T .

The sets in T are called the open sets of the topological space (X, T ).(1) The usual topology TR on R:

U ∈ TR ⇐⇒ U = ∅ or union of open intervals (a, b).

(2) X 6= ∅ =⇒ trivial topology Ttrivial := ∅, X, and discrete topologyTdiscrete := 2X . Note that Ttrivial ⊂ Tdiscrete.

1.3. TOPOLOGY 11

(3) Let T1, T2 be two topologies on X. We say T1 is coarser than T2 or T2 isfiner then T1, if T1 ⊂ T2. Then, for any topology T on a nonempty setX, we have Ttrivial ⊂ T ⊂ Tdiscrete.

Let (X, T ) be a topological space.(1) A neighborhood of x (resp. of A) in X is a set N(x) (resp. N(A)) contain-

ing an open set which contains x (resp. A).

Theorem 1.3.1. Let (X, T ) be a topological space. A subset A ⊂ X is open⇐⇒ it is aneighborhood of each of A.

♣ Exercise: Proof Theorem 1.3.1.

(2) x ∈ X is a limit point of A ⊂ X if ∀ neighborhood N(x) of x contains atleast one point a ∈ A different from x.

Theorem 1.3.2. A ⊂ X is closed⇐⇒ A contains all its limit points.


(3) The closure of A ⊂ X:

A := A ∪ limit points of A.(4) The support of f : X → K:

supp( f ) := x ∈ X : f (x) 6= 0.(5) The interior of A ⊂ X:

A := the largest open set contained in A.

(6) A is dense in X if A = X.(7) A is nowhere dense in X if A has an empty interior.(8) X is separable if it has a countable dense subsets.

1.3.2. Separation. A topological space (X, T ) is Hausdorff if any two distinctpoints possess disjoint neighborhoods.

(1) In a Hausdorff space the points are closed subsets.(2) The usual topology on R and the discrete topology are Hausdorff.(3) The trivial topology is not Hausdorff.(4) A topological space is normal if it is Hausdorff and if any disjoint closed

sets F1 and F2 have disjoint open neighborhoods U1 and U2.

1.3.3. Base. A base B for a topology T is a subsystem of T which satisfieseither one the following equivalent conditions:

(i) ∀ element of T is the union of elements of B;(ii) ∀ x ∈ X and ∀ U ∈ T with x ∈ U ∃ B ∈ B such that x ∈ B ⊂ U.

Then T is said to be generated by B and is denoted TB .


(1) A base for the usual topology of R:

B = (a, b) : a, b ∈ R.

(2) A base for neighborhoods of x ∈ X is a family N of neighborhoods of xsuch that any neighborhood of x contains a member of N .

(3) The first countable space is a topological space in which each point hasa countable base of neighborhoods.

(4) The second countable space is a topological space which topology is gen-erated by a countable base B.

1.3.4. Convergence. A sequence (xn)n≥1 of points in a topological space (X, T )is a mapping N→ X given by n 7→ xn.

(1) A sequence (xn)n≥1 converges to x ∈ X if ∀ neighborhood N of x ∃ n0 ≥ 1such that N contains all xn (∀ n ≥ n0). Write limn→∞ xn = x or xn → x.

(2) If (X, T ) is a first countable space then the topology can be described interms of sequences.

Let (I,≤) be a directed set and (X, T ) a topological space.(1) A net (xα)α∈I in X is a mapping I → X given by α 7→ xα.(2) A net (xα)α∈I converges to x ∈ X if ∀ neighborhood N of x ∃ α0 ∈ I such

that xα ∈ N for all α ≥ α0.(3) Let (I′,≤′) be another directed set. A subnet (yα′)α′∈I′ of a net (xα)α∈I

is a mapping I′ → X given by α′ 7→ xφ(α′) = yα′ , where φ : I′ → I isa mapping such that ∀ α ∈ I ∃ α′ ∈ I′ for which φ(β′) ≥ α wheneverα′ ≤′ β′.

1.3.5. Covering and compactness. A system (Ui)i∈I of (resp. open) subsets ofa topological space (X, T ) is a (resp. an open) covering if ∀ x ∈ X ∃ Ui such thatx ∈ Ui (i.e., ∪i∈IUi = X).

(1) U = (Ui)i∈I is finite if |I| < +∞.(2) A subcovering of the covering U is a subset of U which is itself a cover-

ing.(3) The covering V = (Vj)j∈J is a refinement of the covering U = (Ui)i∈I if∀ j ∈ J ∃ i ∈ I such that Vj ⊂ Ui.

(4) A covering U = (Ui)i∈I is locally finite if ∀ x ∈ X ∃ neighborhood Nsuch that N ∩Ui1 ∩ · · · ∩Uik 6= ∅ for some finite indices i1, · · · , ik ∈ I.

(5) A ⊂ X is compact if it is Hausdorff (relative to the subspace topology)and if any open covering of A has a finite open subcovering.

Theorem 1.3.3. (1) A compact subspace of a Hausdorff space is necessarily closed.(2) Any closed subspace of a compact space is compact.(3) (Bolzano-Weierstrass) A Hausdorff space is compact⇐⇒ ∀ net has a conver-

gent subnet.(4) In a metric space, a set A ⊂ X is compact ⇐⇒ ∀ sequence in A contains a

convergent subsequence with limit in A.(5) (Heine-Borel theorem) The compact subsets of Rn are closed bounded subsets

of Rn.

1.3. TOPOLOGY 13

The Heine-Borel theorem is in general not true.

A ⊂ X is relatively compact if A ⊂ X is compact. A space is locally compactif any point has a compact neighborhood. A Hausdorff space is paracompact ifany open covering has a locally finite refinement.

Theorem 1.3.4. All metric spaces are paracompact. In general, any locally compact,second countable, Hausdorff space is paracompact.

1.3.6. Connectedness. A topological space (X, T ) is disconnected if ∃ dis-joint nonempty open subsets A and B of X such that A ∪ B = X. Otherwise, X issaid to be connected.

Theorem 1.3.5. A topological space X is connected if and only if the only subsets whichare both open and closed are ∅ and X.

A topological space X is locally connected if any neighborhood of any x con-tains a connected neighborhood.

1.3.7. Continuous mappings. Let f : X → Y be a mapping between twotopological spaces (X, TX) and (Y, TY). We say that f is continuous at x ∈ X ifany neighborhoods B ⊂ Y of y = f (x) ∃ neighborhood A of x such that f (A) ⊂ B.

(1) f is continuous on X or is TX/TY-continuous if it is continuous at allx ∈ X.

Theorem 1.3.6. f : X → Y is continuous⇐⇒ ∀ open U in Y, V := f−1(U) is open inX.

Theorem 1.3.7. f : X → Y is continuous ⇐⇒ the net ( f (xα))α∈I converges to f (x)whenever the net (xα)α∈I converges to x.

♣ Exercise: Proof Theorem 1.3.6 and Theorem 1.3.7.

(2) f is continuous =⇒ the sequence ( f (xn))n≥1 converges to f (x) wheneverthe sequence (xn)n≥1 converges to x.

(3) The converse of (2) is not true for non-countable spaces.

Theorem 1.3.8. (i) The image of a continuous mapping of a compact space is compact.(ii) Any continuous function on a compact space takes on its minimum and maximum.


(4) A continuous mapping f : X → Y is proper if f−1(A) is compact when-ever A is compact in Y.

(5) f : (X, TX)→ (Y, TY) is continuous =⇒ ∀ T ′X ⊃ TX and ∀ T ′Y ⊂ TY, wesee that f is also T ′X/T ′T-continuous.

(X, TX)f−−−−−−→

continuous(Y, TY)

finer

y ycoarser

(X, discrete topology)f−−−−−−→

continuous(Y, trivial topology)

(6) We can ask the following question:

(X, ??? the coarsest topology)f−−−−−−→

continuous(Y, ??? the finest topology)

Let X be a set and (Xα)α∈A a family of topological spaces.(6.1) The projective topology on X with respect to (Xα, fα)α∈A is the coars-

est topology on X for which each fα is continuous:

fα : X −→ Xα.

(6.2) The inductive topology on X with respect to (Xα, gα)α∈A is the finesttopology on X for which each gα is continuous:

gα : Xα −→ X.

(7) A homeomorphism is a bijection f which is bi-continuous (i.e., f and f−1

are continuous).

Theorem 1.3.9. A continuous bijection f : X → Y between two topological spaces is ahomeomorphism in the following cases:

(i) X and Y are compact, or(ii) (Banach) X and Y are Banach spaces and f is linear.

1.3.8. Simply-connectedness and path-connectedness. A covering space ofa topological space X is a pair (X, f ) where X is a connected and locally connectedspace, and f is a continuous mapping of X onto X such that ∀ x ∈ X ∃ neighbor-hood V of x satisfying that the restriction of f onto each connected component Cα

of f−1(V) is a homeomorphism from Cα onto V.

(1) Two covering spaces (X1, f1) and (X2, f2) are isomorphic if X1 is homeo-morphic to X2 by ϕ and f2 = f1 ϕ−1:

X1ϕ //

f1 AAA

AAAA

A X2

f2

X

(2) X is simply-connected if X is connected and locally connected, and anycovering space (X, f ) is isomorphic to the trivial covering space (X, IdX).

1.3. TOPOLOGY 15

(3) R is simply-connected and (R, π) is a covering space of R := R/Z,where π(x) := x + Z.

(4) (X, f ) is an universal covering space for X if it is a covering space of Xand X is simply-connected.

(5) X is locally simply-connected if ∀ x ∈ X has a simply-connected neigh-borhood.

Theorem 1.3.10. (i) A connected and locally connected space has a universal coveringspace.

(ii) If a topological space X admits a universal covering space, it admits only one, upto isomorphisms.

(6) Let (X, f ) be a universal covering space of a topological space X. Thefundamental group of X is defined to be

π1(X) = ϕ : ϕ : X → X homeomorphic and f ϕ = f .

By Theorem 1.3.9 the fundamental group is independent of the choice ofX. Moreover, π1(X) is indeed a group.

(7) X is path-connected if ∀ a, b ∈ X ∃ continuous path between them, thatis, ∃ continuous mapping γ : [0, 1]→ X such that γ(0) = a and γ(1) = b.

(8) X is locally path-connected if ∀ x ∈ X and ∀ neighborhood V of x ∃neighborhood U ⊂ X which is path-connected.

path-connected/locally path-connected =⇒ connected/locally connected.

However the converse is not true in general: the set

D :=(x, y) ∈ R2 : y = sin

1x

, x > 0

is locally path-connected, but it not path-connected.

♣ Exercise: Proof the above fact.

Let X be a topological space. Two paths γ0 and γ1 in X are homotopic if ∃continuous mapping

F : [0, 1]× [0, 1] −→ X, (t, s) 7−→ γs(t) := F(t, s)

such that F(t, 0) = γ0(t) and F(t, 1) = γ1(t).

Theorem 1.3.11. If X is a path-connected and locally path-connected space, and anyclosed path in X is homotopic to a constant, then X is simply-connected.

As a consequence, π1(Sn) is simply-connected for any n ≥ 2, but, π1(S

1) ∼= Z.The Poincare conjecture states that any closed simply-connected three-dimensionalmanifold is diffeomorphic to S3. This conjecture is completely solved now by us-ing the Ricci flow, where Hamilton, Perelman, and lots of people make great work.


1.3.9. Associated topologies. Let (X, T ), (X1, T1), (X2, T2) be topologicalspaces.

(1) The relative topology on A ⊂ X is

TA := A ∩U : U ∈ T .(2) (A, T ′A) is a topological space and A ⊂ X. We say that A ⊂ X is a

topological inclusion, written as A → X, if T ′A ⊃ TA.(3) A base for the product topology on X1 × X2 is U1 ×U2 : Ui ∈ Ti, 1 ≤

i ≤ 2. The α-th projection mapping is

πα : X1 × X2 −→ Xα, (x1, x2) 7−→ xα, 1 ≤ α ≤ 2.

The product topology is the coarsest topology in which each πα is contin-uous.

(4) All the open balls on Rn form a base for the usual topology on Rn that isequivalent to the product topology on R× · · · ×R (n times).

Let (Xα)α∈A be a family of topological spaces.(1) Cartesian product:

∏α∈A

Xα :=

x : A −→

⋃α∈A

Xα : xα = x(α) ∈ Xα

.

(2) Letπα : ∏

α∈AXα −→ Xα, x 7−→ xα.

(3) The product topology on ∏α∈A Xα is the coarsest topology in which πα

is continuous. A base for the product topology consists of

∏1≤i≤n

Uαi × ∏α 6=i1,··· ,in

Xα, Uαi ∈ Tαi .

(4) Tychonoff’ theorem: The arbitrary product of compact spaces is compactwith respect to the product topology.

1.3.10. Topology related on other structures. A set X together with a groupoperation an a topology is said to be a topological group if

X× X −→ X, (x, y) 7−→ xy and X −→ X, x 7−→ x−1

are both continuous.

A topological space X which is also a vector space on K is said to be a topo-logical vector space if

X× X −→ X, (x, y) 7−→ x + y and K× X −→ X, (λ, x) 7−→ λx

are both continuous.(1) Rn together with its usual topology is a topological vector space.(2) A topological vector space is said to be locally convex if it admits a base

of neighborhood of 0 (zero element) made of convex sets.

Theorem 1.3.12. Let T : X → Y be a linear mapping between two topological spaces.Then T is continuous if and only if T is continuous at 0 ∈ X.

1.3. TOPOLOGY 17

(3) If X and Y are topological vector spaces, then we denote by L(X, Y) theset of all linear continuous mappings from X to Y, and

(1.3.1) X′ ≡ topological dual of X := L(X, K).

For any x′ ∈ X′ write x′(x) = 〈x′, x〉 ∈ K for any x ∈ X.(4) A bounded set in a topological vector space is a set which can be mapped

inside any neighborhood of the origin by a homothetic transformation ofsufficiently small ratio centered on the origin (x 7→ εx).

1.3.11. Metric spaces. A metric space is a pair (X, d), where X is a set andd : X× X → R is a mapping, satisfying

(i) d(x, y) ≥ 0,(ii) d(x, y) = 0⇐⇒ x = y,

(iii) d(x, y) = d(y, x),(iv) d(x, z) ≤ d(x, y) + d(y, z).

We call d(x, y) the distance between x and y.(1) Open balls:

B(x, r) := y ∈ X : d(x, y) < r.

Theorem 1.3.13. (X, d) is a metric space =⇒ TB is a topology on X, where B =B(x, r) : x ∈ X, r > 0.

(2) TB is called the topology induced by the metric d.(3) (X, d) is a metric space =⇒ X is first countable, Hausdorff and normal.(4) Examples.

(4.1) (X, d0) is a metric space =⇒ discrete topology, where d0(x, y) = 0 ifx = y, and 1 if x 6= y.

(4.2) X = R2, define

d1(x, y) := [(x1 − y1)2 + (x2 − y2)2]1/2,

d2(x, y) := max|x1 − y1|, |x2 − y2|,d3(x, y) := |x1 − y1|+ |x2 − y2|.

(4.3) (Rn, d) is a metric space, with

d(x, y) :=

(∑

1≤i≤n(xi − yi)2

)1/2

.

Euclidean topology on Rn induced by d is equivalent to the producttopology on R× · · · ×R (n times).

(4.4) Let (X, d) be a metric space. For x, y ∈ X define

Path(x, y) := γ : [0, 1]→ X|γ continuous and γ(0) = x, γ(1) = y.Then

dx,y(γ1, γ2) := sup0≤t≤1

d(γ1(t), γ2(t))

is a metric on Path(x, y).


(4.5) Let (X, d) be a metric space and γ : [0, 1]→ X a path. Define

[γ] :=

γ′ : [0, 1]→ X continuous∣∣∣ γ′ = γ ϕ, ϕ : [0, 1]→ [0, 1]

monotonic and continuous

.

Then (X, d) is a metric space, where

X := [γ]|γ : [0, 1]→ X continuous

and

d([γ1], [γ2]) := infγ′i∈[γi ], 1≤i≤2

(sup

0≤t≤1d(γ′1(t), γ′2(t))

).

♣ Exercise: Verify (4.1) – (4.5).

Theorem 1.3.14. Let f : X → Y be a mapping from a metric space (X, d) to a topologicalspace (Y, T ). Then f is continuous at x if and only if ( f (xn))n≥1 converges to f (x)whenever (xn)n≥1 converges to x.

(5) A Cauchy sequence in a metric space (X, d) is a sequence (xn)n≥1 suchthat limn→∞ d(xn, xm) = 0.

(6) A metric space is complete if any Cauchy sequence is convergent.(7) Any metric space is dense in a complete metric space.

A topological invariant is a property of topological spaces which is preservedunder a homeomorphism. For example, separation properties, compactness, con-nectedness, etc.

Exercise 1.3.15. Show that the completeness is not a topologically invariant prop-erty.

A topological vector space X is said to be metrizable if its topology can beinduced by some metric d invariant by translation, i.e.,

d(x, y) = d(x + z, y + z), ∀ x, y, z ∈ X.

A Frechet space is a complete, metrizable, topological vector space.

1.3.12. Banach spaces. Let X be a vector space over K. A norm on X is amapping || · || : X → R such that

(i) ||x + y|| ≤ ||x||+ ||y||,(ii) ||λx|| = |λ|||x||,

(iii) ||x|| = 0⇐⇒ x = 0.Then ||0|| = 0, ||x|| ≥ 0, and

||x− y|| ≥∣∣∣∣||x|| − ||y||∣∣∣∣.

1.3. TOPOLOGY 19

♣ Exercise: Please check it!

(1) || · || is a seminorm if it satisfies (i) and (2).(2) A norm || · || induces a metric on X which is invariant by translation:

||x− y|| := d(x, y).

(3) A normed vector space is a pair (X, || · ||) where X is a vector space and|| · || is a norm on X. Therefore

(X, || · ||) −−−−→ (X, d) −−−−→ (X, T )∥∥∥ ∥∥∥ ∥∥∥normed vector space metric space topological space

(4) || · || and || · ||′ are norms on X such that ||x| < ||x||′ for all x ∈ X =⇒B′(x, ε) ⊂ B(x, ε) and T ⊂ T ′.

(5) || · || and || · ||′ are norms on X and T ⊂ T ′ =⇒ ∃ λ > 0 such that||x|| < λ||x||′ for all x ∈ X.

Indeed, the open ball Bε := B(0, ε) contains B′η := B′(0, η) =⇒ ηx/2||x||′ ∈B′η ⊂ Bε =⇒ ||ηx|| < 2ε||x||′ =⇒ λ = 2ε/η > 0.

Proposition 1.3.16. Any normed vector space must be a metrizable topological vectorspace.

♣ Exercise: Proof Proposition 1.3.16.

Proposition 1.3.17. Any normed vector space must be a locally convex topological vectorspace.

PROOF. A base B = B(x, r) : x ∈ X, r > 0 for (X, || · ||). We shall provethat B(x, r) is convex. ∀ x1, x2 ∈ B(x, r) and λ ∈ [0, 1], ||λx1 + (1− λ)x2 − x|| ≤λr + (1− λ)r = r.

(6) A seminormed vector space is a pair (X, || · ||) where X is a vector spaceand || · || is a seminorm. It is clear that Proposition 1.3.17 is also true forany seminormed vector spaces.

(7) A Banach space is a complete normed vector space. According to Propo-sition 1.3.16 any Banach space is Frechet.

Example 1.3.18. (1) (Rn, || · ||) is complete, where

||x|| := d(0, x) =

(∑

1≤i≤n(xi)2

)1/2

.


(2) (CB(X), || · ||CB(X)) is complete, where(1.3.2)

CB(X) := f : X → R continuous and bounded, || f ||CB(X) := supx∈X| f (x)|.

(3) If U ⊂ Rn is open, then (CkB(U), || · ||Ck

B(U)) is complete, where

CkB(U) :=

f : U → R

∣∣∣∣ continuous and uniformly boundedderivatives of order ≤ k

,

and

(1.3.3) || f ||CkB(U) := sup

x∈U∑|α|≤k|Dα f (x)|

with

α := (α1, · · · , αn) ∈Nn, |α| = ∑1≤i≤n

αi, Dα f (x) := ∏1≤i≤n

(∂

∂xi

)αi

f (x).

(4) If U ⊂ Rn is open, then (L2(U), || · ||L2(U)) is not complete, where

(1.3.4) L2(U) := f : U → R continuous and square integrableand

|| f ||L2(U) :=[∫

U| f (x)|2dx

]1/2.

(8) Compact subsets of a metric space are also closed and bounded.

Let (X, || · ||) be a normed space. A sequence ( fn)n≥1 converges strongly tof ∈ X if

limn→∞

|| f − fn|| = 0.

Consequently, limn→∞ || fn|| = || f ||.

Let (X, T ) be a topological vector space, and X′ := L(X, K) be its topologicaldual.

(1) ( fn)n≥1 converges weakly to f ∈ X, written as fn f , if

limn→∞〈g, fn〉 = 〈g, f 〉, ∀ g ∈ X′.

(2) If T is induced from a norm || · || on X, then (over R)(2.1) fn f =⇒ || f || ≤ lim infn→∞ || fn||,(2.2) fn f and limn→∞ || fn|| = || f || =⇒ limn→∞ || fn − f || = 0.

(3) ( fn)n≥1 ⊂ X′ converges ∗-weakly to f ∈ X′, written as fn ∗ f , if

limn→∞〈 fn, g〉 = 〈 f , g〉, ∀ g ∈ X.

(4) If X is reflextive (i.e., X′ = X), then weak topology is equivalent to ∗-weak topology.

1.4. MEASURES AND INTEGRATIONS 21

Theorem 1.3.19. All bounded subsets are relatively compact in the weak topology of areflextive Banach space.

1.3.13. Hilbert spaces. A Hilbert space is a complete pre-Hilbert space (H , || ·||), where ||x|| := (x|x)1/2.

(1) A Hilbert space is real if the underlying vector space is defined on R.

Theorem 1.3.20. (Riesz’s representative theorem) There is an isomorphism betweena Hilbert space H and its topological dual H ′ defined by

H −→H ′, x 7−→ (x|·).

(2) A subset (xα)α∈A of a Hilbert space H is called an orthonormal base ofH if it is a base such that

(xα|xβ) = δαβ, ∀ α, β ∈ A.

(3) Any Hilbert space H admits an orthonormal base.(3.1) Two orthonormal bases have the same cardinality.(3.2) Two Hilbert spaces with orthonormal bases of the same cardinality

are isomorphic.(4) Any separable Hilbert space is isomorphic to L2((0, 1)).(5) For any element x of a separable Hilbert space we have

x = ∑n≥1

(x|xn)xn,

where (xn)n≥1 is a countable base of H .

1.4. Measures and integrations

1.4.1. Measures. Let X be a nonempty set.(1) A ring of subsets of X is a nonempty class R of subsets of X such that

A, B ∈ R =⇒ A \ B, A ∪ B ∈ R.

Hence φ ∈ R and R is closed under finite unions and finite intersections.(2) A ring of subsets R is a ring under

A + B := A4B = (A \ B) ∪ (B \ A) ∈ R,AB := A ∩ B = (A ∪ B) \ (A4B) ∈ R.

Moreover the zero element and the unit element are respectively ∅ andX.

(3) The ring R is called a field if X ∈ R. It is then denoted A .(4) The ring R (resp. the field A ) is called a σ-ring (resp. σ-field) if R (resp.

A ) is closed under countable unions:

Ai ∈ Ri (i ≥ 1) =⇒⋃i≥1

Ai ∈ R.


(5) A σ-field A can equally defined by the following axioms:

A ∈ A =⇒ X \ A ∈ A , Ai ∈ A =⇒⋃i≥1

Ai ∈ A .

(6) Let E be a class of subsets of X =⇒ ∃! σ-field A (E ), called the σ-fieldgenerated by E , containing E :

(1.4.1) A (E ) := the smallest class of subsets of Xcontaining E such that A (E ) is a σ-field.

The existence is guaranteed by Zorn’s lemma.(7) A measurable space is a pair (X, A ), where X is a set and A is a σ-field.

Let (X, T ) be a topological space.(1) The Borel σ-field of X is the σ-field generated by the open sets of X (or

equivalently the closed sets of X). An element of the Borel σ-field is calleda Borel set.

(2) ∪i≥1Ui, ∪i≥1Fi, ∩i≥1Ui, ∩i≥1Fi are Borel sets, where Ui and Fi are respec-tively open and closed sets.

(3) Gδ-set: ∩i≥1Ui (Ui open), Fσ-set: ∪i≥1Fi (Fi closed), Gδσ-set: ∪i≥1Gi (Giare Gδ-sets), Fσδ-set: ∩i≥1Fi (Fi are Fσ-sets).

(4) The Borel σ-field of R is generated by the open intervals (a, b) or equiv-alently by the closed intervals [a, b], or the semi-closed [a, b) or (a, b] or(−∞, a] or (−∞, a) or [a,+∞) or (a,+∞).

Indeed, an open set is a countable union of open intervals. For exam-ple

(a, b) =⋃

n≥1

[a +

1n

, b)

, [a, b) =⋃

n≥1

[a, b− 1

n

].

1.4.2. Measure spaces. Let (X, T ) be s topological space.(1) A positive set function on X is a mapping m : A → [0,+∞], where

φ ∈ A ⊂ 2X .(1.1) m is finitely additive if ∀ disjoint finite family (Ai)1≤i≤n of A with

∪1≤i≤n Ai ∈ A ,

m

( ⋃1≤i≤n

Ai

)= ∑

1≤i≤nm(Ai), m(∅) = 0.

(1.2) m is countably additive if ∀ disjoint countable family (Ai)i≥1 of Awith ∪i≥1 Ai ∈ A ,

m

(⋃i≥1

Ai

)= ∑

i≥1m(Ai), m(∅) = 0.

(1.3) m is σ-finite if X = ∪i≥1 Ai with m(Ai) < +∞.(1.4) m is finite if X ∈ A and m(X) < +∞.

(2) Let (X, A ) be a measurable space.(2.1) A positive measure m on (X, A ) is a countably additive, positive set

function m : A → [0,+∞].


(2.2) A measure space is a triple (X, A ,m), where (X, A ) is a measurablespace and m is a positive measure on (X, A ).The elements of A are the measurable subsets of X, and m(A) is themeasure of A ∈ A .

Theorem 1.4.1. (Hahn extension theorem) Let X be a set, Σ a field of subsets of X,and µ : Σ → [0,+∞] a countably additive positive set function. Then there is a positivemeasure µ on the σ-field generated by Σ such that µ Σ = µ and µ is unique when µ isσ-finite.

Σµ //

inclusion

[0,+∞]

A (Σ)µ

::uuuuuuuuu

(3) A probability space is a measure space (X, A ,m) with m(X) = 1. Theelements A of A are called events and m(A) is the probability of A.

(4) A property is said to hold m-almost everywhere (m-a.e.) if it holds for allpoints of X except possibly for points of a set A of measure m(A) = 0.

(5) A positive measure m in the measure space (X, A ,m) is said to be com-plete if ∀ A ∈ A with m(A) = 0 and ∀ B ⊂ A, we have B ∈ A andm(B) = 0.

Any measure space (X, A ,m) can be included into a complete mea-sure space (X, A ,m).

(6) A positive measure m on (X, B), where X is a locally compact Hausdorffspace and B is the Borel σ-field, is called a Borel measure if the measureof every compact set (so is closed and hence measurable) is bounded.

(7) Let Σ be a field of subsets of X and µ : Σ → [0,+∞] be finitely additive,positive set function. We say that µ is regular if ∀ A ∈ Σ and ∀ ε > 0 ∃F, G ∈ Σ such that

F ⊂ A ⊂ G and µ(G \ F) < ε.

Theorem 1.4.2. (Alexandroff theorem) Let X be compact topological space, Σ a fieldof subsets of X. If µ is a finite, regular, finitely additive, positive set function on Σ, then µis countably finite.

(9) A Borel measure µ is regular⇐⇒ ∀ Borel set A and ∀ ε > 0 ∃ closed setF and open set U such that F ⊂ A ⊂ U and µ(U \ F) < ε.

Let f : X → Y be a set mapping.(1) Given a metric space (X, A ,m), define the image by f of (X, A ,m) to be

the metric space (Y, B, n), where

B := B ∈ 2Y : f−1(B) ∈ A , n(B) := m( f−1(B)) (∀ B ∈ B).

(2) A positive measure (A ,m) on X is invariant by f : X → X if

f−1(A) ∈ A (∀ A ∈ A ) and m(A) = m( f−1(A)).


Let M = (X, A ,m) and M′ = (X′, A ′,m′) be two σ-finite measure spaces.(1) Define the σ-field A ⊗A ′ of subsets of X× X′ by

A ⊗A ′ := the smallest σ-field containing all the sets ofthe form A× A′, where A ∈ A and A′ ∈ A ′

(2) By Theorem 1.4.1, ∃! positive measure m⊗ m′ defined on A ⊗A ′ suchthat

(m⊗m′)(A× A′) = m(A)m′(A′), ∀ (A, A′) ∈ A×A ′.

(3) The product measure space:

M×M′ = (X× X′, A ⊗A ′,m⊗m′).

The Lebesgue measure on Rn.(1) The Lebesgue measure on R:

(R, B, l),

where R is the real line with its usual topology, B is the Borel σ-field gen-erated by open intervals, and l, called the Lebesgue measure on R, is theonly regular Borel measure, up to multiplication by constant, invariantby translation.

(2) Define l as follows:(2.1) l((a, b)) := b − a, ∀ open interval (a, b) ∈ R (=⇒ l is an additive

positive set function).(2.2) Extend l, defined in (2.1), to the field generated by open intervals

=⇒l([a, b]) = l([a, b)) = l((a, b]) = b− a.

(2.3) If [a, b) = ∪i≥1[ai, bi), where [ai, bi) are pairwise disjoint, then

l([a, b)) = ∑i≥1

l([ai, bi)).

(3) Cantor set:C := [0, 1] \ S,

where

S :=⋃i≥1

Si, S1 =

(13

,23

), S2 =

(19

,29

)∪(

79

,89

), · · · .

Then C is uncountable, but is measure zero with respect to l.(4) The Lebesgue measure on Rn:

Rn :=

R× · · · ×R︸︷︷︸n

, B ⊗ · · · ⊗B︸︷︷︸n

, l⊗ · · · ⊗ l︸︷︷︸n

.


Let G be a locally compact topological group and B its Borel σ-field.(1) A (left) Haar measure µ is a left-invariant (i.e., invariant under the map-

pings G → G, x 7→ gx, that is, µ(gA) = µ(A) for all Borel sets A and allg ∈ G), regular, Borel measure on G. Similarly, we can define right Haarmeasure µ (that is µ(Ag = µ(A)).

Theorem 1.4.3. There is a unique Haar measure, modulo a constant factor, on a givenlocally compact topological group.

(2) Let G = R∗ = R \ 0 with group law the usual multiplication. Thendx/|x| is a Haar measure:∫

Rf (tx)

dx|x| =

∫R

f (x)dx|x|

for all f ∈ L1(G, l).


1.4.3. Measurable functions. Let f : X → Y be a mapping between two mea-surable spaces (X, A ) and (Y, B).

(1) f is measurable or A /B-measurable if f−1(B) ∈ A for all B ∈ B.(2) Examples:

(2.1) If (Y, B, n) is the image of (X, A ,m) by f : X → Y, then f is measur-able.

(2.2) If (X, A ) and (Y , B) are topological spaces with Borel σ-fields, thenany continuous mapping from X to Y is measurable.

(3) The composition of two measurable mappings is also measurable:

(X, A )f−−−−−−→

measurable(Y, B)

g−−−−−−→measurable

(Z, C ) =⇒ g f measurable.

Then we obtain a category MS:

Ob(MS) : measurable spaces,

and

HomMS((X, A ), (Y, B)) = measurable mappings.(4) Let (X, A ,m) be a measure space and f : X → R a real function. We

say f is measurable if it is a measurable mapping from (X, A ) to R, orequivalently, if x ∈ X : a < f (x) < b ∈ A , ∀ a < b.

(4.1) A complex-valued function f +√−1g is measurable if both f and g

are measurable.(4.2) f and g are measurable functions on X and λ ∈ K =⇒ λ f , f + g,

f − g, | f | are measurable.(5) (X, A ,m) is a complete measurable space, f = g is m-a.e. =⇒ f is mea-

surable if and only if g is measurable.


Theorem 1.4.4. (Lusin) Let X be a locally compact topological space with X = ∪i≥1Xi(Xi being compact), and m a Borel measure on X. TFAE:

(i) a function f on X is measurable,(ii) ∀ compact set K ⊂ X and ∀ ε > 0 ∃ compact subset Kε ⊂ K and gε ∈ C(Kε, R)

such thatgε = f |Kε and m(K \ Kε) < ε.

This function gε may be extended to a continuous function g with compact sup-port in X, and supX | f | = supX |g|.

Let (X, A ,m) be a measure space.(1) If ( fn)n≥1 is a sequence of measurable functions on (X, A ,m) that are

finite m-a.e., then we say that ( fn)n≥1 converge in measure to the mea-surable function f , if ∀ ε > 0 one has

limn→+∞

m (x ∈ X : | fn(x)− f (x)| > ε) = 0.

Notion: fn →m f .(2) We have the following relations:

convergence in measure a.e.

×y×

pointwise convergence a.e.

convergence in measure a.e.xµ(X)<+∞

pointwise convergence a.e.

Theorem 1.4.5. (Egorov) Let (X, A ,m) be a measure space with m(X) < +∞. If( fn)n≥1 converges pointwise a.e. to a finite measurable function f , then ∀ ε > 0 ∃Aε ⊂ X such that m(X \ Aε) < ε and ( fn)n≥1 uniformly converges to f on Aε.

1.4.4. Integrable functions. Let (X, A ,m) be a measure space.(1) A function on (X, A ,m) is said to be simple if it is zero except on a finite

number n of disjoint sets Ai ∈ A of finite measure m(Ai), where thefunctions is equal to a finite constant ki (1 ≤ i ≤ n).

(2) If f is a simple function, we define

(1.4.2)∫

Xf dm := ∑

1≤i≤nkim(Ai).

(3) Any measurable function f : X → R := R ∪ ±∞ ∃ ( fn)n≥1 such thatfn are simple functions and fn → f pointwise. If moreover f ≥ 0, thenthe fn may be chosen positive and ( fn)n≥1 increasing

0 ≤ f1 ≤ · · · ≤ fn ≤ fn+1 −→ f .

(4) For any measurable function f : X → [0,+∞], define

(1.4.3)∫

Xf dm := sup

∫X

ρdm : 0 ≤ ρ ≤ f , ρ simple

.

We then say that f is integrable if∫

X f dm is finite.


Theorem 1.4.6. (Monotone convergence theorem) Let (X, A ,m) be a measure spaceand ( fn : X → [0,+∞])n≥1 an increasing integrable sequence which is convergent m-a.e.to f . Then

limn→∞

∫X

fn dm exists =⇒ f is integrable and∫

Xf dm = lim

n→∞

∫X

fn dm.

Theorem 1.4.7. (Fatou) Let (X, A ,m) be a measure space and ( fn : X → [0,+∞])n≥1an measurable sequence which is convergent m-a.e. to f . Then

(1.4.4)∫

Xf dm ≤ lim inf

n→∞

∫X

fn dm.

(5) For ∀ f : X → R we have

f = f+ − f−with f± : X → [0,+∞], and say f+ (resp. f−) the positive part (resp.negative part) of f . We say f is integrable if f± are both integrable. Itsintegral is defined to be∫

Xf dm :=

∫X

f+ dm−∫

Xf− dm.

Theorem 1.4.8. Let (X, A ,m) be a measure space. A measurable function f : X → R isintegrable if and only if | f | : X → [0,+∞] is integrable.

(6) Integration on A ∈ A . If A = ∪i≥1 Ai ∈ A with (Ai)i≥1 disjoint, then

(1.4.5)∫

Af dm =

∫X

f χA dm = ∑i≥1

∫Ai

f dm

for any measurable function f : A→ R.(7) Basic properties:

(7.1) For any λ, µ ∈ R,∫A(λ f + µg)dm = λ

∫A

f dm+ µ∫

Agdm.

(7.2) | f | ≤ |g|, g integrable, f measurable =⇒ f integrable.(7.3) f ≤ g m-a.e., f and g integrable =⇒∫

Af dm ≤

∫A

gdm.

(7.4) f measurable, | f | bounded on A ∈ A with m(A) < +∞ =⇒ f isintegrable and ∣∣∣∣∫A

f dm∣∣∣∣ ≤ Mm(A)

where | f | ≤ M on A.


(7.5) f ≥ 0, A ⊂ B and A, B ∈ A =⇒∫A

f dm ≤∫

Bf dm.

Theorem 1.4.9. (Lebesgue dominated convergence theorem) Let (X, A ,m) be ameasure space and ( fn)n≥1 an integrable sequence, fn → f a.e., | fn| ≤ g with g beingintegrable. Then f is integrable and

(1.4.6) limn→∞

∫X

fn dm =∫

Xlim

n→∞fn dm =

∫X

f dm.

(9) For (X, A ,m) = (X1 × X2, A1 ⊗A2,m1 ⊗m2), define∫X1×X2

f dm1dm2 :=∫

Xf dm.

Theorem 1.4.10. (Fubini) A measurable function f : X1 × X2 → R is integrable⇐⇒one of the following integrals∫

X1

[∫X2

| f |dm2

]dm1 and

∫X2

[∫X1

| f |dm1

]dm2

exists and is finite.If f is integrable, then

(1.4.7)∫

X1×X2

f dm1dm2 =∫

X1

[∫X2

f dm2

]dm1 =

∫X2

[∫X1

f dm1

]dm2.

(10) Let (Y, B, n) be an image under u of a measure space (X, A ,m).(10.1) A measurable function f on Y is integrable on Y ⇐⇒ f u is inte-

grable on X. Then

(1.4.8)∫

Yf dn =

∫X

f udm =∫

X(u∗ f )dm.

(10.2) f is measurable on Y =⇒ f u is measurable on X.(11) Let (X, A ,m) be a measure space and Y ∈ A . Then (Y, A ,m) is a mea-

sure space, where

A :=

A = Y ∩ A : A ∈ A

, m(A) := m(A).

We say (A ,m) the measure induced by m on Y.If ι : Y → X denotes the inclusion, then for any measurable function

f on X such that– χY f is measurable on X,– χY f is m-integrable,– f ι is m-integrable,

we obtain ∫Y

f ιdm =∫

XχY f dm =

∫Y

f dm.


1.4.5. Integration on locally compact spaces. Let (X, B,m) be a measure space,where X is a locally compact topological space, B is the Borel σ-field, and m is aBorel measure.

(1) m(compact set) is finite.(2) ∀ continuous function f with compact support =⇒∫

Xf dm is finite.

(3) The space of continuous functions with compact support on X is densein the space of integrable functions on X.

Lebesgue integral.(1) (Rn, BRn , ln) =⇒ dx := dln and∫

Rnf dln =

∫Rn

f (x)dx.

Theorem 1.4.11. (Change of variable) Suppose that f is Lebesgue integrable on anopen subset V ⊂ Rn and ϕ : U → V, x 7→ y = ϕ(x), is a diffeomorphism. Then

(1.4.9)∫

Vf (y)dy =

∫U( f ϕ)(x)

∣∣det[ϕ′(u)]∣∣ dx

where ϕ′(x) = (∂yi/∂xj) = D(y)/D(x).

(2) Examples:(2.1) ∃ Lebesgue integrable but NO Riemann integrable:

f : [0, 1] −→ R, x 7−→ f (x) =

1, x ∈ Q,0, x /∈ Q.

(2.2) ∃ functions on R which are not Lebesgue integrable but have finiteimproper Riemann integrals (a measurable function f is Lebesgueintegrable if and only if | f | is Lebesgue integrable):

sin xx

, cos(x2), sin(x2), · · · .

(2.3) @ reasonable definition of improper Riemann integrals on Rn:

I :=∫∫

x,y>0sin(x2 + y2)dxdy.

The domain D = (x, y) ∈ R2 : x, y > 0 can be approximated byD′n := [0, n]× [0, n] or D′′n := x2 + y2 ≤ n2 : x, y > 0. However

I′n :=∫∫

D′nsin(x2 + y2)dxdy→ π

4, I′′n =

∫∫D′′n

sin(x2 + y2)dxdy =1− cos(n2)

2.

Radom measure. Let X be a locally compact space.


(1) A Radom measure µ on X is a continuous linear form

µ : D0(X) −→ R

where D0(X) is the space of continuous functions with compact supporton X endowed with the inductive limit topology of the topologies of uni-form convergence on compact sets.

(2) µ is continuous⇐⇒ ∀ compact set K ⊂ X ∃ C(K) > 0 such that ∀ contin-uous function f with support in K,

|µ( f )| ≤ C(K) supK| f |.

(3) A Radom measure µ is positive if µ( f ) ≥ 0 for all 0 ≤ f ∈ D0(X).(4) To any Borel measure m on X, we can associate a positive Radom measure

µ defined by

(1.4.10) µ( f ) :=∫

Xf dm, f ∈ D0(X).

Theorem 1.4.12. (Riesz-Markov) Suppose that X is a locally compact space and X =∪i≥1Xi with Xi being compact. Then ∀ positive Radom measure µ on X ∃! regular Borealmeasure m such that (1.4.10) holds for all f ∈ D0(X).

1.4.6. Signed and complex measures. A signed measure space is a triple(X, A ,m), where (X, A ) is a measurable space and m : A → (−∞,+∞] is acountably additive set function.

Theorem 1.4.13. (Jordan decomposition theorem) Given a signed measure space(X, A ,m), there is a unique decomposition

m = m+ −m−

where• m± are positive measures.• ∃ A ∈ A such that m−(A) = 0 and m+(X \ A) = 0.

The positive measure (A , |m|) with |m| := m+ + m− is called the total variation of(A ,m).

(1) f is integrable with respect to m⇐⇒ f is integrable with respect to |m|.(2) f is integral with respect to m =⇒∫

Xf dm :=

∫X

f dm+ −∫

Xf dm−,

∣∣∣∣∫Xf dm

∣∣∣∣ ≤ ∫X| f |d|m|.

A complex measure space is a triple (X, A ,m), where (X, A ) is a measurablespace and m : A → (−∞,+∞] +

√−1(−∞,+∞] is a set function with m = m1 +√

−1m2 and mi being signed measures.


(1) The total variation (A , |m|) of (A ,m) is defined to be(1.4.11)

|m|(A) := sup

∑i≥1|m(Ai)| : (Ai)i≥1 pairwise disjoint in A and ∪i≥1 Ai ⊆ A

.

Then

(1.4.12) |m(A)| ≤ |m|(A).

(2) If m is a signed measure, then |m|(A) defined in (1) agrees with its defi-nition in terms of the Jordan decomposition.

(3) Let f : X → C, f = f1 +√−1 f2, and f1, f2 real. We say that f is inte-

grable with respect to the complex measure m = m1 +√−1m2, if fi is

integrable with respect to mi (1 ≤ i ≤ 2). Its integral is∫X

f dm :=∫

Xf1 dm1 −

∫X

f2 dm2 +√−1(∫

Xf1 dm2 +

∫X

f2 dm1

).

(4) f is m-integrable⇐⇒ | f | is |m|-integrable and f is measurable. Moreover∣∣∣∣∫Xf dm

∣∣∣∣ ≤ ∫X| f |d|m|.

1.4.7. Integration of vector-valued functions. Let f be a mapping from ameasure space (X, A ,m) to a Banach space (E, B, || · ||).

(1) The first difficulty: f and g measurable ; f ± g is measurable.(2) The second difficulty: f measurable ; ∃ ( fn)n≥1 simply functions con-

verging pointwise to f .(3) When (X, A ,m) is a σ-finite measure space and (E, B, || · ||) is a separable

Banach space, the above two difficulties do not arise.(4) We say f is a simple mapping if it has value zero except on a finite

number Ai (1 ≤ i ≤ n) of subsets Ai ∈ A of X, with finite measurem(Ai) < +∞, where it has a constant value in E:

f = ∑1≤i≤n

aiχAi , ai ∈ E.

(5) For any simple mapping f , define∫X

f dm := ∑1≤i≤n

aim(Ai) ∈ E.

(6) A Cauchy sequence of simple mappings is a sequence ( fn)n≥1 of simplemappings such that ∀ ε > 0 ∃ N ∈N such that∫

X|| fn − fk||dm < ε

for all n, k > N.(6.1) ∀ n ≥ 1, one has∣∣∣∣∣∣∣∣∫X

fn dm∣∣∣∣∣∣∣∣ ≤ ∑

1≤i≤Mn

||ai,n||m(Ai,n) ≤∫

X|| fn||dm.

(6.2) ( fn)n≥1 Cauchy sequence of simple mappings =⇒ (∫

X fndm)n≥1 is aCauchy sequence in E =⇒ limn→∞

∫X fndm exists in E.


(7) f : X → E is m-integrable if ∃ Cauchy sequence ( fn)n≥1 of simple map-pings, converging a.e. to f . The integral of f is∫

Xf dm := lim

n→∞

∫X

fn dm

which is independent of the choice of ( fn)n≥1.

Theorem 1.4.14. Let f : (X, A ,m) → (B, B, || · ||) be a measurable mapping from ameasure space into a Banach space. Then

f is m-integrable ⇐⇒ || f || : X → [0,+∞) is integrable.

Then ∣∣∣∣∣∣∣∣∫Xf dm

∣∣∣∣∣∣∣∣ ≤ ∫X|| f ||dm.

1.4.8. L1-spaces. Let (X, A ,m) be a measure space.(1) Define

L 1(X) := integrable functions over X

=

f : X → R∪ +∞

∣∣∣∣ f measurable and∫X | f |dm < +∞

.(1.4.13)

(2) Define

L 1(X) −→ R, f 7−→∫

X| f |dm.

It is linear, subadditive, and positive homogeneous. Moreover,∫X| f |dm = 0 =⇒ f = 0 m− a.e.

(3) We say f and g is L 1(X) are equivalent if

f ∼ g f = g m− a.e.

Let

(1.4.14) L1(X) := L 1(X)/ ∼=[ f ] : f ∈ L 1(X)

.

Then

L1(X) −→ R, [ f ] 7−→∫

X| f |dm = || f ||L1(X)

is well-defined. Thus

L1(X) ∼=space of (classes of) functions defined a.e.

and integrable on X, together with thenorm f 7→ || f ||L1(X) =

∫X | f |dm.

Theorem 1.4.15. (Riesz-Fischer) If (X, A ,m) is a measure space, then L1(X) is a Ba-nach space.


Theorem 1.4.16. If (X, A ,m) is a measure space, where X is a locally compact topologicalspace, A the Borel σ-field, and m a Borel measure, then D0(X), the space of continuousfunctions with compact support, is dense in L1(X).

(4) Theorem 1.4.15 and Theorem 1.4.16 hold when f takes its values in aBanach space (E, B, || · ||).

1.4.9. Lp-spaces. Let (X, A ,m) be a measure space.

(1) Set

(1.4.15) Lp(X) =the space of (classes of) measurable functionsdefined a.e. on X, such that | f |p is integrable.

and

(1.4.16) || f ||Lp(X) :=[∫

X| f |pdm

]1/p=

[∣∣∣∣∣∣∣∣| f |p∣∣∣∣∣∣∣∣L1(X)

]1/p

.

(2) The Minkowski inequality:

(1.4.17) || f + g||Lp(X) ≤ || f ||Lp(X) + ||g||Lp(X), p ≥ 1.

The Holder inequality:

(1.4.18) || f g||L1(X) ≤ | f ||Lp(X)||g||L1(X),1p+

1q= 1, p ≥ 1.

(3) p ≥ 1 =⇒ (Lp(X), || · ||Lp(X)) is a normed space.(4) On L2(X):

(1.4.19) ( f |g)L2(X) =∫

Xf gdm, ( f | f )L2(X) = || f ||2L2(X).

(5) In the space Lp(X) strong convergence is also called convergence in themean of order p.

(6) The space Lp(X) (p ≥ 1 or p = ∞) are locally convex.(7) (Lp(X), || · ||) (p ≥ 1) are Banach spaces, and (L2(X), (·|·)L2(X)) is a Hilbert

space.

Theorem 1.4.17. (Riesz) For each p ≥ 1, the space Lp(X) is complete, that is, ∃ f ∈Lp(X) such that

limn→∞

|| fn − f ||Lp(X) = 0

if limm,n→∞ || fn − fm||Lp(X) = 0.

(8) C2([a, b]) is NOT complete.



Theorem 1.4.18. (Inclusion theorem) X is of finite measure (e.g., X is a compact subsetof Rn together with the Lebesgue measure) =⇒

1 ≤ p′ < p =⇒ Lp(X) ⊂ Lp′(X).

Actually,

|| f ||Lp′ (X)≤ || f ||Lp(X)[m(X)]

1p′ −

1p .

(9) Let

L∞(X) =the space of (class of a.e. defined) measurable function bounded

almost everywhere on X which has the norm given by|| f ||L∞(X) := infM > 0 : | f (x)| ≤ M a.e. on X

(10) ∀ g ∈ Lq(X) (1 ≤ q ≤ +∞), define g′ ∈ (Lp(X))′ ( 1p + 1

q = 1) by

g′ : Lp(X) −→ C, f 7−→∫

Xf gdm.

Hence

Lq(X) ⊂ (Lp(X))′, 1 ≤ p, q ≤ +∞ and1p+

1q= 1.

Theorem 1.4.19. (Riesz representation theorem) 1 < p < +∞ and 1/p + 1/q = 1=⇒ (Lp(X))′ = Lq(X). In particular, (L2(X))′ = L2(X).

(11) If (X, A ,m) is a measure space and m is σ-finite, then

(L1(X))′ ∼= L∞(X).

(12) In general, it is not true that (L∞(X))′ is L1(X). For example, consider(X, A ,m) = (R, B, l) and x0 ∈ R. Define

L : CB(R) −→ R, f 7−→ f (x0).

Then L is linear and continuous in the L∞(R) norm,

|L( f )| = | f (x0)| ≤ || f ||L∞(R).

By the Hahn-Banach extension theorem, ∃ continuous linear functionalon L∞(R) such that the restriction to CB(R) is L. But this linear formcannot be given by

L( f ) =∫

Rf gdx

for some g ∈ L1(R). For instance, the Dirac measure cannot be repre-sented by a locally integrable function.


Exercise 1.4.20. Let (X, A ,m) be a measure space. The decreasing rearrangementof a measurable function f defined on X is the function f ∗ : [0,+∞)→ R by

f ∗(t) := infs > −0 : d f (s) ≤ t,where d f : [0,+∞)→ R denotes the distribution function of f and is given by

d f (s) := m (x ∈ X : | f (x)| > α) .

Show that(1) d f+g(α + β) ≤ d f (α) + dg(β) and d f g(αβ) ≤ d f (α) + dg(β) for any mea-

surable functions f , g and any α, β > 0.(2) For f ∈ Lp(X), 0 < p < +∞, we have

|| f ||pLp(X)= p

∫ ∞

0αp−1d f (α)dα.

(3) On (Rn, B, ln = dx) compute d f and f ∗ for

f (x) =1

1 + |x|p , 0 < p < +∞.

(4) Show that f ∗(d f (α)) ≤ α whenever α > 0 and d f ( f ∗(t)) ≤ t whenevert > 0, where d is a measurable function.

(5) ( f + g)∗(t1 + t2) ≤ f ∗(t1) + f ∗(t2) and ( f g)∗(t1 + t2) ≤ f ∗(t1)g∗(t2) formeasurable functions f , g and any t1, t2 > 0.

(6) d f = d f ∗ and∫X| f |p dm =

∫ ∞

0| f ∗(t)|pdt, 0 < p < +∞,

for any measurable function f .Given 0 < p, q ≤ +∞ and a measurable function f define

|| f ||Lp,q(X) :=(∫ +∞

0

(t1/p f ∗(t)

)q dtt

)1/q, q < +∞

and|| f ||Lp,∞(X) := sup

t>0

(t1/p f ∗(t)

), q = +∞.

The Lorentz space is defined to be

Lp,q(X) :=The space of (classes of) measurable functionsdefined a.e. on X, such that || f ||Lp,q(X) < +∞

Note that Lp,p(X) = Lp(X) and L∞,∞(X) = L∞(X). Compute(7) If f = ∑1≤j≤N ajχEj , where the sets Ej have finite measure and are

pairqise disjoint and a1 > · · · > aN , then

|| f ||Lp,q(X) =

(pq

)1/q [aq

1Bq/p + aq2(Bq/p

2 − Bq/p1 ) + · · ·+ aq

N(Bq/pN − Bq/p

N−1)]1/q

when 0 < p, q < +∞, and

|| f ||Lp,∞(X) = sup1≤j≤N

(ajB

1/pj

)


when q = +∞. Here Bj := ∑1≤i≤j m(Ei).(8) For 0 < p < +∞ and 0 < q ≤ +∞ we have

|| f ||Lp,q(X) = p1/q(∫ +∞

0

[d f (s)1/ps

]q dss

)1/q.

A subset A of (X, A ,m) id called an atom if m(A) > 0 and every subset Bof A has measure either equal to zero or equal to m(A). We say that (X, A ,m) isnon-atomic if it contains no atoms. Equivalently, (X, A ,m) is non-atomic if andonly if ∀ A ∈ A with m(A) > 0 ∃ proper subset B ( A in A with m(B) > 0 andm(A \ B) > 0. (Rn, B, ln) is non-atomic. It can be shown that if (X, A ,m) is anon-atomic σ-finite measure space, then

(L1,q(X))′ = L∞(X) (0 < q ≤ 1) =⇒ (L1(X))′ = L∞(X),

and

(Lp,q(X))′ = Lp′ ,q′(X) (1 < p, q < +∞, 1/p + 1/p′ = 1 = 1/q + 1/q′).

1.5. Linear functional analysis

An operator on a vector space X is a mapping T : X → Y from a subset of X,called the domain Dom(T) of T, onto a subset of another vector space Y, calledthe range Range(T) of T.

1.5.1. Bounded linear operators. A continuous linear operator T on a vectorspace X can be extended to a continuous linear operator with domain X.

(1) Let

L(X, Y) := continuous linear operators from X to Y.(2) A linear operator T : X → Y between normed spaces is bounded if ∃

K ≥ 0 such that

||T(x)||Y ≤ K||x||X , ∀ x ∈ X.

Theorem 1.5.1. A linear operator T : X → Y between normed spaces is continuous ifand only if T is bounded.

(3) Let T : X → Y be a continuous linear operator between normed spaces.Define the norm of T to be

(1.5.1) ||T|| := infK > 0 : ||T(x)||Y ≤ K||x||X = sup||T(x)||Y : ||x||X = 1.(4) An unbounded operator: X = (C∞([0, 1]), || · ||L∞) with || f ||L∞ := sup[0,1] | f |,

T := d/dx. Then T is linear but not bounded. Since

||T(sin kt)||L∞ = k = k|| sin kt||L∞

for all k > π/2.(5) If X is a finite-dimensional normed vector space, then ∀ linear operator

is continuous and hence bounded.

1.5. LINEAR FUNCTIONAL ANALYSIS 37

Theorem 1.5.2. If X is a normed space and Y is a Banach space, then (L(X, Y), || · ||) isBanach.

(6) X normed space =⇒ X′ := L(X, K) Banach space =⇒ bidual X′′ :=L(X′, K) Banach space.

Theorem 1.5.3. If X is a normed space, then there exists an isomorphism J : X →J(X) ⊂ X′′.

PROOF. Define

J : X −→ X′′, x 7−→(〈·, x〉 : X′ −→ K, x′ 7−→ 〈x′, x〉

).

Then J is linear, injective, and is an isometry.

(7) A normed space X is reflective if X′′ = X.(8) ∀ 1 < p < +∞ =⇒ Lp(X) is reflective.

A Banach algebra is a Banach space together with an associative internal op-eration (called multiplication), (B, ·) or B.

(1) X Banach space =⇒B(X) := L(X, X) is a Banach algebra with

(T1T2)x := T1(T2x) =⇒ ||T1T2|| ≤ ||T1|| · ||T2||.(2) An involutive Banach algebra is a Banach algebra with a norm preserv-

ing involution ∗: (B,+, ·, || · ||, ∗). An involution ∗ in B is a mappingB→ B given by T 7→ T∗ (adjoint of T) such that

(2.1) (T + S)∗ = T∗ + S∗,(2.2) (αT)∗ = αT∗,(2.3) (ST)∗ = T∗S∗,(2.4) T∗∗ = T,(2.5) ||T∗|| = ||T||.T ∈ B is self-adjoint (resp. normal) if T = T∗ (resp. TT∗ = T∗T).

(3) A C∗-algebra is an involutive Banach algebra which in addition satisfies(2.6) ||T∗T|| = ||T||2.

(4) Examples:(4.1) In C, the mapping z 7→ z is an involution.(4.2) X Hilbert space =⇒ canonical involution on B(X) is (T f |g) = ( f |T∗g).

Let X be a Banach space.

(1) A linear operator T on X has the transposed operator T on X′ defined by⟨Tx′, y

⟩= 〈x′, Ty〉, x′ ∈ X′, y ∈ X.

(2) X Hilbert space =⇒ (T)∗ = T∗.

Let X be a Banach space and T ∈ B(X).


(1) λ ∈ C is said to be in the spectrum σ(T) if T− λI is not bijective.(1.1) If T− λI is not injective (i.e., Tx− λx = 0 has a solution x 6= 0), λ is

called an eigenvalue of T, and is said to be in the point spectrum ofT. The corresponding solutions are called eigenvectors. The set ofall point spectrums of T is denoted σp(T).

(1.2) If T− λI is injective but not surjective.(1.2.1) λ is said to be in the continuous spectrum of T, if Range(T−

λI) is dense in X. The set of all continuous spectrums of T isdenoted σc(T).

(1.2.2) Otherwise, λ is said to be in the residual spectrum of T. Theset of all residual spectrums of T is denoted σr(T).

Hence

(1.5.2) σ(T) = σp(T) t σc(T) t σr(T).

(2) dim(X) < +∞ =⇒ σ(T) = σp(T).

Theorem 1.5.4. X Banach space =⇒ ∀ T ∈ B(X) has the following properties:

σ(T) 6= ∅ and σ(T) ⊂ D||T|| ⊂ C.

Theorem 1.5.5. X Hilbert space, T ∈ B(X) self-adjoint =⇒(i) σr(T) = ∅.

(ii) σ(T) ⊂ R and ||T|| = supλ∈σ(T) |λ|.(iii) Eigenvectors corresponding to distinct eigenvalues are orthogonal.

Theorem 1.5.6. X Banach space =⇒(i) T ∈ B(X) with ||T− I|| < 1 is invertible and its inverse is

(1.5.3) T−1 = I + ∑n≥1

(I− T)n.

(ii) Breg(X) := invertible elements in B(X) is open in B(X).(iii) Breg(X)→ Breg(X), T 7→ T−1, is homeomorphic.

1.5.2. Compact operators. Let f : X → Y be a mapping between two metricspaces (X, dX) and (Y, dY).

(1) f is compact if(1.1) f is continuous, and(1.2) f maps every bounded subset of X into a relatively compact subset

in Y.(2) If X, Y are Hausdorff topological vector spaces and f : X → Y is linear

and maps every bounded subset of X into a relatively compact subset ofY, then f is compact.

1.5. LINEAR FUNCTIONAL ANALYSIS 39

(3) Let F be a family of functions defined on (X, dX). We say F is equi-continuous if ∀ ε > 0 ∃ δ > 0 such that

| f (x)− f (x′)| < ε

whenever x, x′ ∈ X with dX(x, x′) < δ and whenever f ∈ F .

Theorem 1.5.7. (Ascoli-Arzela theorem) X compact metric space, C(X) space of con-tinuous functions on X with the uniform norm =⇒∀ bounded and equi-continuous subsetK ⊂ C(X) is compact.

Let (X, dX) be a compact metric space, and (Y, dY) a complete metric space.(1) F , a family of mappings from X to Y, is equicontinuous if ∀ ε > 0 ∃

δ > 0 such thatdY( f (x), f (x′)) < ε

whenever x, x′ ∈ X with d)X(x, x′) < δ and whenever f ∈ F .(2) Theorem 1.5.7 is still valid, if “K is bounded” is replaced by “∀ x ∈ X, f (x) : f ∈ K is relatively compact in Y”.

(3) Example:(3.1) Let (X, dX) be a compact metric space, (Y, dY) be a compact metric

space with a regular Borel measure n, and K : X×Y → R be contin-uous. Define

T : C(Y) −→ C(X), f 7−→ (T f )(x) :=∫

YK(x, ·) f dn.

Then T is linear, continuous, and compact, because for any boundedsubset K of C(Y), T(K) is a bounded, equicontinuous subset of C(X)and then is compact.

(3.2) Let (X, A ,m) and (Y, B, n) be measure spaces. Define

T : L2(Y) −→ L2(X), f 7−→∫

YK(x, ·) f dn,

where K ∈ L2(X × Y). Then T is a linear compact operator. Also Tis a Hilbert-Schmidt operator, i.e., if (en)n≥1 is a base of L2(Y), then

∑n≥1||Tei||L2(X) < +∞.

Theorem 1.5.8. (Fredholm alternative) X Banach space, 0 6= λ ∈ C, T ∈ B(X)linear compact operator =⇒ one and only one of the two following statements is true:

(i) T f − λ f = g has one solution f for each g ∈ X (i.e., T− λI is isomorphic);(ii) T f − λ f = 0 has no zero solution (i.e., λ is an eigenvalue of T). For each λ

except possibly λ = 0, the solutions span a finite dimensional subspace of X.

Theorem 1.5.9. (Riesz-Schauder) The spectrum of a compact operator has no accumu-lation point other than, possibly, zero.


Theorem 1.5.10. (Adjoint theorem) T f − λ f = 0 has a nonzero solution⇐⇒ T f −λ f = 0 has a nonzero solution.

Let λα be an eigenvalue of T. T f − λα f = g⇐⇒ g is such that 〈 fα, g〉 = 0 for allfα satisfying T fα − λα fα = 0.

Theorem 1.5.11. (Hilbert-Schmidt) Let H be a Hilbert space and T a self-adjoint com-pact operator. Then ∃ orthonormal base of H made of eigenvectors of T.

1.5.3. Open mapping and closed graph theorems. Over Banach spaces, thereare three fundamental theorems: uniform boundedness theorem, open mappingtheorem, and closed graph theorem.

Theorem 1.5.12. (Uniform boundedness theorem) Let Tn : X → Y be a sequenceof linear continuous mappings between Frechet spaces. If (||Tnx||Y)n≥1 is uniformlybounded (i.e., independent of n) for each x ∈ X, then (||Tn||)n≥1 is also uniformlybounded.

In particular, if limn→∞ Tnx exists for each x ∈ X, then the limit Tx :=limn→∞ Tnx is linear and continuous.

Theorem 1.5.13. (Open mapping theorem) If T : X → Y is a linear continuoussurjective mapping between Frechet spaces, then T is open.

Corollary 1.5.14. (Banach theorem) If T : X → Y is a linear continuous bijectivemapping between Frechet spaces, then T is isomorphic.

Let T : X → Y be a mapping between metric spaces.

(1) T is closed if ∀ (xn)n≥1 ⊂ Dom(T) with dX(xn, x) → 0 for some x ∈ X,and (Txn)n≥1 converges to y ∈ Y, then x ∈ Dom(T) and Tx = y.

(2) The graph of (T, Dom(T)) is the subset of X × Y with elements (x, Tx),∀ x ∈ Dom(T).

(3) T is closed⇐⇒ Graph(T) is closed in X×Y.

Theorem 1.5.15. (Closed graph theorem) If T : X → Y is a closed linear map withdomain X between Frechet spaces, then T is continuous.

1.6. DIFFERENTIABLE CALCULUS ON BANACH SPACES 41

1.6. Differentiable calculus on Banach spaces

Let (X, || · ||X) and (Y, || · ||Y) be Banach spaces, U ⊂ X open, x0 ∈ U, andf : X → Y a mapping.

(1) f is said to be differentiable at x0 if ∃ Df|x0 ∈ L(X, Y) such that

(1.6.1) f(x0 + h)− f(x0) = Df|x0 h + R(h)

where x0, x0 + h ∈ U, f(x0), f(x0 + h), R(h) ∈ Y, and ||R(h)||Y = o(||h||X).We call

Df|x0 = Df(x0) = f′(x0) = Dx0 fthe Frechet differential at x0 or derivative at x0.

(2) The differential of f : X → K is called a functional derivative.(3) f is differentiable in U if it is differentiable at every x0 ∈ U. The differ-

ential Df is a mapping U → L(X, Y) given by x 7→ Df|x.(4) If Df is continuous, f is said to be continuously differentiable or of class

C1.Examples:(1) f : U ⊂ R→ R is differentiable =⇒ D f |x0 = d f /dx|x0 ∈ R.(2) L : X → Y linear mapping =⇒ DL|x0 = L.(3) C : X → Y constant mapping =⇒ DC|x0 = 0.(4) f : R→ X =⇒ Df|t = f′(t) ∈ L(R, X), h 7→ f′(t)h. Then

f′(t)h = h[f′(t)1], ∀ h ∈ R.

Here f′1 : t 7→ f′(t)1 ∈ X can be identified with f′ : R→ X.(5) f : Rn → Rp, x 7→ y = f(x), where

yα = f α(x1, · · · , xn), 1 ≤ α ≤ p with f α ∈ C1.

The Jacobian matrix of f at x0 is

Df|x0 h =

(∑

1≤i≤n

∂ f α

∂xi

∣∣∣∣x0

hi

)1≤α≤p

=(

∂i f α|x0 hi)

1≤α≤p.

When X, Y are non-Banach, locally convex topological vector spaces, we sayf : U → Y, U open in X, is differential at x0 ∈ U, if

f(x0 + h)− f(x0) = Df|x0 h + R(h), h ∈ X,

where x0, x0 + h ∈ U, f(x0), f(x0 + h), R(h) ∈ Y, Df|x0 ∈ L(X, Y), and R is tangentto zero.

(1) Let o(t) be a real function of t ∈ R∩ (−1, 1) such that limt→0 o(t)/t = 0.(2) R is said to be tangent to zero if ∀ neighborhood N of 0Y ∈ Y ∃ neighbor-

hood V of 0X ∈ X such that R(tV) ⊂ o(t)N.

Theorem 1.6.1. (Composite mapping) Let X, Y, Z be Banach spaces, U ⊂ X open,V ⊂ Y open, f : U → Y differential at x0 , g : V → Z differential at y0 = f(x0),x0 ∈ U, and y0 ∈ V =⇒ h := g f is differential at x0 and

(1.6.2) Dh(x0) = Dg(y0) Df(x0).


Theorem 1.6.2. (Inverse mapping) U ⊂ X open, V ⊂ Y open, f : U → V invertibleand differential at x0 ∈ U, and Df(x0) : X → Y is an isomorphism =⇒ f−1 is differentialat y0 = f(x0) ∈ V and

(1.6.3) Df−1(y0) =1

Df(x0).

PROOF. f is differential at x0 =⇒ f(x0 + h) − f(x0) = Df(x0)h + R(h). Lety0 + k = f(x0 + h). Then

(y0 + k)− y0 = Df(x0)h + R(h)

and hence[Df(x0)]

−1(k) = h + [Df(x0)]−1(R(h)).

Therefore

f−1(y0 + k)− f−1(y0) = h = [Df(x0)]−1k− [Df(x0)]

−1(R(h)).

We need to check that

lim||k||Y→0

||[Df(x0)]−1(R(h))||X||k||Y

= 0.

Because lim||h||X ||R(h)||Y/||h||X = 0, we have ||R(h)||Y ≤ c||h||X for some c > 0with 0 < 1− c||[Df(x0)]

−1|| < 1. According to∣∣∣∣∣∣[Df(x0)]−1(k)

∣∣∣∣∣∣X=∣∣∣∣∣∣h + [Df(x0)]

−1(R(h))∣∣∣∣∣∣

X≥ ||h||X−

∣∣∣∣∣∣[Df(x0)]−1(R(h))

∣∣∣∣∣∣X

≥(

1− c∣∣∣∣∣∣[Df(x0)]

−1∣∣∣∣∣∣) ||h||X

so that lim||k||Y→0 ||h||X/||k||Y = 0. From

||[Df(x0)]−1(R(h))||X||k||Y

=||[Df(x0)]

−2(R(h))||X||h||X

· ||h||X||k||Ywe obtain the desired limit.

1.6.1. Diffeomorphisms. A mapping f : U → V is said to be diffeomorphicif f is a bijection with f and f−1 continuously differentiable (of class C1).

(1) homomorphism + of class C1 ; diffeomorphism ( f (x) = x3).

Theorem 1.6.3. A homeomorphism f : U → V of class C1 is a diffeomorphism ⇐⇒Df(x) is an isomorphism for every x ∈ U.

♣ Exercise: Proof Theorem 1.6.3. (Hint: Use Theorem 1.6.2)

(2) X = ∏1≤i≤n Xi product of Banach spaces, Y Banach space, U ⊂ X open,f : U → Y a mapping, and a given point a = (a1, · · · , an).


(2.1) The partially constant mapping ei : Xi → X, where

ei(xi) := (a1, · · · , ai−1, xi, ai+1, · · · , an).

Then

ei(xi + hi)− ei(xi) = (0, · · · , 0, hi, 0, · · · , 0)

so

Dei : Xi −→ X, hi 7−→ (0, · · · , 0, hi, 0, · · · , 0) =: hi · (0, · · · , 0, 1, 0, ·, 0)

called the partially identity mapping. We can view Dei as a map-ping Xi → X given by above.

(2.2) The partial derivative of f : U → Y at a is

∂f∂xi

∣∣∣∣a≡ f ′xi (a) := D(f ei)

∣∣∣∣ai= Df(a) Dei|ai = Df(a) Dei(ai).

(2.3) Moreover

∑1≤i≤nb

f′xi (a)hi = ∑1≤i≤n

(Df(a) Dei(ai)

)hi = Df(a)h.

(2.4) The existence of f′ ⇐⇒ the existence of partial derivatives; but theconverse is not true.

(2.5) f′xi : U → L(Xi, Y), x 7→ f′xi (x) =⇒ existence and continuity of Df ifand only if existence and continuity of f′xi .

(2.6) Examples:(2.6.1) f : Rn → Rp, x = (x1, · · · , xn) 7→ f(x) = ( f α(x))1≤α≤p =⇒

f′xi (x0) = (∂ f α/∂xi(x0))1≤α≤p ∈ Rp

(2.6.2) U ⊂ Cm(Rn) open, V ⊂ C(Rn) open, and P : U → V, u 7→P(Dmu), nonlinear partial differential operator of order m, where

Dmu := derivatives of u of order ≤ m,P := corresponding function C1 in all its arguments.

Then P is differential at u0 ∈ U and

DP(u0) : Cm(Rn) −→ C(Rn), h 7−→ ∑|j|≤m

∂P∂Dju

(Dmu0)Djh,

the linearization of P that is a linear partial differential opera-tor of order m.

Theorem 1.6.4. (The mean value theorem) X, Y Banach spaces, U ⊂ X convex opensubset, f : U → V of class C1 =⇒ ∀ x ∈ U and ∀ x + h ∈ U, we have

f(x + h)− f(x) =∫ 1

0Df(x + th)hdt.

PROOF. Let γ : [0, 1]→ U be defined by γ(t) := x + th ∈ U (U is convex) andu = f γ : [0, 1]→ Y. Then Du(t) = Df(x + th)γ′(t) = Df(x + th)h ∈ Y.


Corollary 1.6.5. U topological space, Y Hausdorff topological space, f : U → Y contin-uous =⇒

(1) f is locally constant and U is connected =⇒ f is constant on U.(2) Df ≡ 0, U is a connected and locally convex topological vector space =⇒ f is

constant on U.

1.6.2. Euler’s equation. Let [a, b] ⊂ R be closed and

L : [a, b]×R×R −→ R, (x, y, z) 7−→ L(x, y, z)

be C1 on R×R.(1) Define

S : C1([a, b]) −→ R, q 7−→∫ b

aL(x, q(x), q′(x))dx

and||q||C1([a,b]) := sup

[a,b](|q|+ |q′|).

(2) Compute, ∀ h ∈ C1([a, b]),

S(q + h) =∫ b

aL(x, q(x) + h(x), q′(x) + h′(x))dx

and

S(q + h)− S(q) =∫ b

a

[h(x)L′y(x, q(x), q′(x)) + h′(x)L′z(x, q(x), q′(x))

]dx

+∫ b

aα(|h(x)|+ |h′(x)|

)dx

where α = o(||h||C1([a,b])). Therefore

DS(q)h =∫ b

a

[h(x)L′y(x, q(x), q′(x)) + h′(x)L′z(x, q(x), q′(x))

]dx.

(3) Assume that L and q are C2 =⇒∫ b

ah′(x)L′z(x, q(x), q′(x))dx

=[h(x)L′z(x, q(x), q′(x))

]ba −

∫ b

ah(x)

ddx[L′z(x, q(x), q′(x))

]dx.

(4) Define

U :=

q ∈ C2([a, b]) : q(a) = q(b) = 0⊂ C2([a, b]).

Then for S : U → R, we have, q ∈ U,

DS(q) : U −→ R, h 7−→ DS(q)h

with

DS(q)h =∫ b

aE (L)hdx


where

E (L) := L′y −d

dxL′x, L := L(x, q(x), q′(x)).

Observe that

DS(q) = 0 ⇐⇒ E (L) = 0. (Euler′sequation)

Euler’s equation for several variables and several unknown functions. Letq : Ω → Rp, x 7−→ qα(x1, · · · , xn), be of class C1, Ω ⊂ Rn be open and bounded,and Dq = (∂qα/∂xi) = (∂iqα).

(1) L : Rnp+p+n → R of class C1.(2) S : ∏p times C1(Ω) −→ R, where

S(q) :=∫

ΩL(x, q(x), Dq(x))dx.

(3) Then

DS(q)h = ∑1≤i≤n, 1≤α≤p

∫Ω

(hαL′qα +

∂hα

∂xi L′∂iqα

)dx.

(4) Assume that L and q are of class C2 =⇒ Define

U :=

q ∈ C2(Ω)× · · · × C2(Ω)︸︷︷︸p times

∣∣∣∣q|∂Ω = 0

.

Then for S : U → R, we have

DS(q)h :=∫

Ω∑

1≤α≤phαEα(L)dx,

whereEα(L) := L′qα − ∑

1≤i≤n

∂

∂xi L′∂iqα .

(5) In general we can replace U in (4) by

Uf :=

q ∈ C2(Ω)× · · · × C2(Ω)︸︷︷︸p times

∣∣∣∣q|∂Ω = f

with the same operator Eα(L). Because q, q + h ∈ Uf implies h|∂Ω = 0.

1.6.3. Higher order differentials. Let (X, || · ||X) and (Y, || · ||Y) be Banachspaces, U ⊂ X be open, and f : U → Y be a C1-mapping.

(1) If Df : U → L(X, Y) is differential, define its second differential of f atx:

(1.6.4) D2f|x ≡ D2f(x) ≡ f′′(x) : X −→ L(X, Y)

and the second variation of f:

(1.6.5) D2f ≡ f′′ ∈ L (U,L(X,L(X, Y))) .

(2) f is of class Cp on U if Dpf exists on U and is continuous.


(3) A diffeomorphism f is of class Cp on U if f and f−1 are of class Cp on U.

Theorem 1.6.6. X, Y, Z Banach spaces =⇒ ∃ natural isomorphism

L(X,L(Y, Z)) −→ L(X×Y, Z)

where

L(X×Y, Z) :=

bilinear continuous mappingsg : X×Y → Z

with norm

||g|| := inf K > 0 : ||g(x, y)||Z ≤ K||x||X ||y||Y .

PROOF. Define

L(X,L(Y, Z)) −→ L(X×Y, Z), f 7−→ α(f)

withα(f) : X×Y −→ Z, (x, y) 7−→ f(x)y.

Conversely, we can define

β(g)(x) := g(x, ·) : Y −→ Z, y 7−→ g(x, y)

for any g ∈ L(X×Y, Z). It can be shown that• ||α(f)|| ≤ ||f||.• ||β(g)|| ≤ ||g||.• α−1 = β.

Therefore, L(X,L(Y, Z)) ∼= L(X×Y, Z).

(4) ∀ x0 ∈ U, we have

f′′(x0) ∈ L(X,L(X, Y)) ∼= L(X× X, Y),(h, k) 7−→ (f′′(x0)h)k =: f′′(x0)(h, k) ∈ Y

Theorem 1.6.7. f : U → Y is twice differential at x0 =⇒ ∀ x ∈ U, f′′(x0) is a bilinearsymmetric mapping.

(5) The bilinear symmetric mapping f′′(x) is called the Hessian of f at x andalso denoted Hessx(f).

(6) When Y = K, we call f ′′(x) the quadratic form of f on X.(7) Examples:

(7.1) Finite-dimensional spaces:

f : Rn −→ Rp, x 7−→ ( f α(x1, · · · , xn))1≤α≤p,

with

f ′′(x0)(h, k) = Hessx0( f )(h, k) =

(∑

1≤i,j≤n

∂2 f α

∂xi∂xj hikj

)1≤α≤p

.


(7.2) Banach-spaces:

S : C1([a, b]) −→ R, q 7−→∫ b

aL(x, q(x), q′(x))dx.

Then

S′′(q) : C1([a, b])× C1([a, b]) −→ R, (h, k) 7−→ S′′(q)(h, k)

with (L is C2 in (y, z))

S′′(q)(h, k) =∫ b

a

[hkL′′yy(x, q(x), q′(x)) + h′k′L′′zz(x, q(x), q′(x))

+ (hk′ + h′k)L′′yz(x, q(x), q′(x))]

dx.

Theorem 1.6.8. (Taylor’s expression with integral remainder) U ⊂ X open, f :U → Y of class Cn, and [x0, x0 + h] ⊂ U =⇒

(1.6.6) f(x0 + h) = f(x0) + ∑1≤k≤n−1

f(k)(x0)

k!hk + R

with

R :=1

(k− 1)!

∫ 1

0(1− t)n−1f(n)(x0 + th)hn dt

wheref(k)(x0) ∈ L(X,L(X, · · · ,L)(X, Y))︸︷︷︸

(k−1) times

andf(k)(x0)hk :=

((f(k)(x0)h)h · · ·

)h︸︷︷︸

k times

.


Corollary 1.6.9. (Lagrange and Peano remainders) U ⊂ X open.(1) f : U → Y of class Cn, [x0, x0 + h] ⊂ U, and f(n)(x0 + th) bounded for all

t ∈ [0, 1] (i.e., ||f(n)(x0 + th)hn||Y ≤ M||h||nX , ∀ t ∈ [0, 1], ∃ M > 0) =⇒

(1.6.7)

∣∣∣∣∣∣∣∣∣∣f(x0 + h)− ∑

0≤k≤n−1

f(k)(x0)

k!hk

∣∣∣∣∣∣∣∣∣∣Y

≤ M||h||nX

n!

(2) f : U → Y of class Cn−1 and f is n differential at x0 =⇒

(1.6.8)

∣∣∣∣∣∣∣∣∣∣f(x0 + h)− ∑

0≤k≤n

f(k)(x0)

k!hk

∣∣∣∣∣∣∣∣∣∣Y

= o(||h||nX).


1.7. Calculus of variations

Let U be open in a Banach space (X, || · ||X) and f : Y → R a mapping.(1) f has a relative minimum at a ∈ U if ∃ neighborhood Va ⊂ U such that

f (x) ≥ f (a) for all x ∈ Va.(2) f has a strictly relative minimum at a ∈ U if ∃ neighborhood Va ⊂ U

such that f (x) > f (a) for all a 6= x ∈ Va.(3) Similarly, we can define relative maximum and strictly relative maxi-

mum.

1.7.1. Necessary conditions for minima. We first prove

Theorem 1.7.1. f is differential at a and f has a relative minimum at a =⇒ f ′(a) = 0.

PROOF. ∃ neighborhood Va ⊂ U such that f (x) ≥ f (a) for all x ∈ Va. Take aball B(a, r) ⊂ Va and consider

g(t) := f (a + th), a + th ∈ B(a, r).

By Fermat’s theorem, 0 = g′(0) = f ′(a)h =⇒ f ′(a) = 0.

Theorem 1.7.2. f is twice differential at a and f has a relative minimum at a =⇒f ′′(a) ≥ 0 (i.e., f ′′(a) : X× X → R is positive, f ′′(a)(h, h) ≥ 0).

PROOF. By Taylor’s theorem,

f (a + h)− f (a) =f ′′(a)

2h2 + ε(h)||h||2X , ε(h) = o(||h||X).

For ||h||X 1 one has

12

f ′′(a)(h, h) + ε(h)||h||2X ≥ 0.

Fix h and choose λ ∈ R with |λ| 1 =⇒12

f ′′(a)(λh, λh) + ε(λh)||λh||2X ≥ 0

and thenf ′′(a)(h, h) + 2ε(λh)||h||2X ≥ 0.

Letting λ→ 0 yields f ′′(a)(h, h) ≥ 0 for all h ∈ X.

1.7.2. Sufficient conditions for minima. U ⊂ X open, f : U → R, and a ∈ U.(1) If f is twice differential at a, then f ′′(a) is said to be non-degenerate if

f ′′(a) : X −→ L(X, R) = X′, h 7−→ f ′′(a)h

is an isomorphism of Banach spaces.(2) f ′′(a) is non-degenerate =⇒ ( f ′′(a)h = 0 ⇐⇒ h = 0). But the converse

only holds when dim X is finite, and in this case f ′′(a) is non-degenerateid and only if det( f ′′xixj) 6= 0.

1.7. CALCULUS OF VARIATIONS 49

Theorem 1.7.3. f ′′(a) non-degenerate and positive =⇒ ∃ λ > 0 such that ∀ h ∈ X,f ′′(a)(h, h) ≥ λ||h||2X .

PROOF. f ′′ non-degenerate =⇒ X → X′, given by h 7→ f ′′(a)h, isomorphic=⇒ ∃ µ > 0 such that

||h||X ≤ µ|| f ′′(a)h||X′ = µ sup| f ′′(a)(h, k)

∣∣||k||X = 1

.

∃ k ∈ X such that ||k||X = 1 and ||h||X ≤ µ| f ′′(a)(h, k)|. f ′′(a) positive =⇒||h||2X ≤ µ2 [ f ′′(a)(h, h)

] [f ′′(a)(k, k)

], ||k||X = 1.

But f ′′(a) continuous =⇒ f ′′(a)(k, k) bounded on ||k||X = 1 =⇒ ∃ M > 0 suchthat ||h||2X ≤ Mµ2[ f ′′(a)(h, h)].

Theorem 1.7.4. (Sufficient conditions) U ⊂ X open and f : U → R of class C2,a ∈ U.

(1) f ′(a) = 0, f ′′(a) positive and non-degenerate =⇒ f has a strict relative mini-mum.

(2) Assume furthermore U is convex.(2.1) f ′(a) = 0 and f ′′|U ≥ 0 =⇒ f has a minimum at a.(2.2) f ′(a) = 0 and f ′′ ≥ 0 in a neighborhood of a =⇒ f has a relative mini-

mum at a.

PROOF. (1) By Taylor’s theorem,

f (a + h) = f (a) +12

f ′′(a)h2 + ε(h)||h||2X , ε(h) = o(||h||2X).

Theorem 1.7.3 =⇒ f ′′(a)(h, h) ≥ λ||h||2X with λ > 0 =⇒

f (a + h) ≥ f (a) +[

λ

2+ ε(h)

]||h||2X .

Taking h sufficiently small so that |ε(h)| < λ/4 yields

f (a + h) ≥ f (a) +λ

4||h||2X > f (a)

for all h 6= 0 and ||h||X 1.(2) Use Theorem 1.6.8 + convexity =⇒

f (x) = f (a) +∫ 1

0

1− t2

f ′′(a + t(x− a))(x− a)2dt ≥ f (a)

for x ∈ U of x in a neighborhood of a.

Example: Let Ω ⊂ Rn be a bounded and open subset. Define

S : C1(Ω) −→ R, q 7−→∫

Ω∑

1≤i≤n

(∂q∂xi

)2dx.


(1) S is differential and

S′ : C1(Ω) −→ L(C1(Ω), R), q 7−→ S′(q)h = ∑1≤i≤n

2∫

Ω

∂q∂xi

∂h∂xi dx.

(2) ∀ q ∈ C2(Ω), S′(q) ∈ L(C1(Ω), R) and

S′(q)h = ∑1≤i≤n

2∫

Ω

[∂

∂xi

(∂q∂xi h

)− ∂2q

(∂xi)2 h]

dx.

(3) ∀ q ∈ C2(Ω) and ∂Ω regular (so that Stokes’ formula can be applied) =⇒

S′(q)h = ∑1≤i≤n

2∫

Ω

[(∂q∂xi h

)niω− ∂2q

(∂xi)2 h]

dx

with niω = (−1)idx1 ∧ · · · ∧ dxi ∧ · · · ∧ dxn.(4) ∀ q ∈ C2(Ω) and ∂Ω regular, q takes given functions on ∂Ω =⇒ h = 0 on

∂Ω and

S′(q)h = −2∫

Ω∑

1≤i≤n

∂2q(∂xi)2 dx.

Hence

∆q0 := ∑1≤i≤n

∂2q0

(∂xi)2 = 0 (Euler′sequation) and S′′(q0)(h, h) ≥ 0,

so that S has a minimum at q0.

1.8. Implicit function theorem and inverse function theorem

Theorem 1.8.1. (Classical theorem) Let f : (x, y) 7→ f (x, y) be a function such that• f (x0, y0) = 0,• f is differential at (x0, y0),• f ′y(x0, y0) 6= 0.

Then f (x, y) = 0 has exactly one continuous solution y = ϕ(x) for x in a neighborhoodof x0 such that ϕ(x0) = y0.

1.8.1. Contracting mapping theorems. (X, d) complete metric space, F ⊂ Xclosed. A contracting mapping is a mapping f : F → F such that

d( f (x), f (y)) ≤ kd(x, y), k ∈ [0, 1), ∀ x, y ∈ F.

We also say that f is Lipschitz of order k < 1.

Theorem 1.8.2. (Contracting mapping theorem) (1) A contracting mapping f hasstrict one fixed point (∃! a ∈ F such that f(a) = a).

(2) (X, || · ||) Banach space, f : B(a, R) → X contracting mapping of ratio k < 1(i.e., ||f(x)− f(y)|| ≤ k||x− y||, ∀ x, y ∈ B(a, R)) =⇒ϕ := 1− f is a homeomorphismof an open set V ⊂ B(a, R) onto B(a − b, (1− k)R) with b = f(a). Moreover ϕ−1 is(1− k)−1-Lipschitz.

1.8. IMPLICIT FUNCTION THEOREM AND INVERSE FUNCTION THEOREM 51

PROOF. (1) Let x0 ∈ F and fn(x0) := f(fn−1(x0)) ∈ F =⇒d( fn(x0), fn−1(x0)) ≤ kd(fn−1(x0), fn−2(x0)) ≤ · · · ≤ kn−1d(f(x0), x0).

k ∈ [0, 1) =⇒ (fn(x0))n≥1 is Cauchy =⇒ ∃ a ∈ F such that

a = limn→∞

fn(x0) = limn→∞

f(fn−1(x0)) = f(a).

If ∃ another b ∈ F with f(b) = b, then

d(a, b) = d(f(a), f(b)) ≤ kd(a, b) =⇒ (1− k)d(a, b) ≤ 0.

Since k < 1, it follows that a = b.(2) ∀ y ∈ B(a− b, (1− k)R) =⇒ the point x such that ϕ(x) = y is unique if it

exists:

||ϕ(x)−ϕ(x′)|| ≥ ||x− x′|| − ||f(x)− f(x′)|| ≥ (1− k)||x− x′||.Let

x0 := a, x1 := f(x0) + y, · · · , xn−1 := f(xn) + y, · · · .Then

• xn ∈ B(a, R) for all n ≥ 0: Assume xn ∈ B(a, R). We now prove xn+1 ∈B(a, R).

||xn+1 − a|| = ||xn+1 − x0|| ≤ ∑0≤i≤n

||xi+1 − xi||

≤ ∑0≤i≤n

ki||y− (b− a)|| ≤ 11− k

||y− (b− a)||.

Because y ∈ B(a− b, (1− k)R) =⇒ ||xn+1 − a|| < R.• ϕ is surjective: The above also shows ||xn+1 − xn|| ≤ kn(1 − k)R =⇒

(xn)n≥1 is Cauchy and its limit x satisfies

x = f(x) + y =⇒ ϕ(x) = y.

Moreover ||x− x′|| ≤ ||ϕ(x)−ϕ(x′)||/(1− k) =⇒ϕ−1 is (1− k)−1-Lipschitz.

1.8.2. Inverse function theorem. We first prove

Lemma 1.8.3. (X, || · ||X), (Y, || · ||Y) Banach spaces =⇒ GL(X, Y) := Isom(X, Y) isopen in L(X, Y).

PROOF. Recall that L(X, Y) is a Banach space with norm ||f|| for any f ∈L(X, Y). Assume GL(X, Y) 6= ∅. Take g0 ∈ GL(X, Y) and choose g ∈ L(X, Y) sothat

||h− 1||L(X,X) < 1, h := g−10 g.

Theorem 1.8.2 =⇒ h ∈ GL(X, X) and

h−1 = ∑n≥0

kn, k := 1− h.

Hence g = g0 h ∈ GL(X, Y) and

g0 ∈ GL(X, Y) =⇒

g ∈ L(X, Y)∣∣∣∣|||g− g0|| <

1||g−1

0 ||


is open in L(X, Y).

Theorem 1.8.4. (Inverse function theorem) (X, || · ||X) and (Y, || · ||Y) Banachspaces, U ⊂ X open, V ⊂ Y open, f : U → V of class C1, a ∈ U, b := f(a) ∈ V,f′(a) : X → Y isomorphism =⇒ ∃ open neighborhoods Ua ⊂ U and Vb ⊂ V such thatf−1 is a C1-diffeomorphism of Vb onto Ua.

PROOF. Let ϕ := 1 − [f′(a)]−1 f : U → X. Then ϕ is C1 in U and ϕ′(a) =1− [f′(a)]−1f′(a) = 1− 1 = 0. Hence ∃ k < 1 such that for R 1

||ϕ′(a + tR)|| < k < 1, ∀ 0 < t < 1.

ϕ is a contracting mapping in the ball B(a, R) =⇒ By Theorem 1.8.2, [f′(a)]−1 f =1 −ϕ is a homeomorphism of U′ ⊂ B(a, R) onto B([f′(a)]−1b, (1 − k)R). Since[f′(a)]−1 is an isomorphism of X onto Y, f is an homeomorphism of U′ onto V′ :=f′(a)B([f′(a)]−1b, (1− k)R). In particular, f is invertible on V′, the equation f(x) =y has strictly one solution x ∈ U′ for y ∈ V′.

Theorem 1.6.2 =⇒ f−1 is differential at b and Df−1(b) = [Df(a)]−1. By Lemma1.8.3, ∃ open neighborhood Ua ⊂ U′ ⊂ U such that f′(x) is an isomorphism for ∀x ∈ Ua. Then by Theorem 1.6.2 again, f−1 is C1-differential on Vb := f(Ua).

1.8.3. Implicit function theorem. We prove

Theorem 1.8.5. (Implicit function theorem) (X, || · ||X), (Y, || · ||Y), (Z, || · ||Z) Ba-nach spaces, U ⊂ X × Y open, f : U → Z of class C1, f(a, b) = 0, f′y(a, b) : Y → Zisomorphism =⇒ ∃ open sets W ⊂ X with a ∈ W and V ⊂ U with (a, b) ∈ V, and aC1-mapping g : W → Y such that

(x, y) ∈ V and f(x, y) = 0 ⇐⇒ y = g(x), x ∈W.

PROOF. SetF : U −→ X× Z, (x, y) 7−→ (x, f(x, y)).

Then

F′ =[

1 0f′x f′y

]and F′(a, b) : X×Y −→ X× Z isomorphism.

Theorem 1.8.4 applied to F(x, y) = (x, f(x, y)) = (x, 0), ∃ open neighborhoods V ⊂U with (a, b) ∈ V and W ⊂ X with a ∈W such that F−1 is a C1-diffeomorphism ofW × 0 onto V =⇒ ∃ g : W → Y that is C1 and y = g(x), ∀ x ∈W.

1.8.4. Global theorems. f : X → Y of class C1 and f′(x) is an isomorphismfor ∀ x ∈ X ; f is a diffeomorphism, even if X = Rn.

Example: Consider

f : R2 −→ R2, (x, y) 7−→ (ex cos y, ex sin y) .

1.8. IMPLICIT FUNCTION THEOREM AND INVERSE FUNCTION THEOREM 53

Figure (Theorem 1.8.7): The segment from b to y0 in Y

Then

f ′(x, y) : R2 −→ R2, f ′(x, y) ∼[

ex cos y −ex sin yex sin y ex cos y

]= ex

[cos y − sin ysin y cos y

].

But f is NOT injective since f (x, y) = f (x, y + 2kπ).

Theorem 1.8.6. (X, || · ||X) and (Y, || · ||Y) Banach spaces, U ⊂ X open, f : U → Y ofclass C1, satisfying

(i) f is injective,(ii) f′(x) : X → Y is isomorphic, ∀ x ∈ U.

Then f is a diffeomorphism from U into f(U) ⊂ Y.

PROOF. By (i), f is a bijection of U onto f(U) ⊂ Y.(1) f(U) is open in Y. Let b ∈ f(U) and a ∈ U be such that f(a) = b. Theorem

1.8.4 =⇒ ∃ open neighborhoods Ua of a in U and Vb of b in Y such that f : Ua → Vbis a diffeomorphism and Vb := f(Ua) ⊂ f(U) ⊂ Y.

(2) Similarly the image by f of any open set in U is open =⇒ f−1 : f(U) → Uis continuous.

(3) f is a C1-homeomorphism of U onto f(U). By Theorem 1.6.2, f is an diffeo-morphism from U into f(U).

Theorem 1.8.7. (X, || · ||X) and (Y, || · ||Y) Banach spaces, f : X → Y of class C1,satisfying

(i) ∀ x ∈ X, f′(x) is an isomorphism,(ii) ∃ M > 0 such that ||[f′(x)]−1|| ≤ M, ∀ x ∈ X.

Then f is a diffeomorphism of X onto Y.

PROOF. According to Theorem 1.8.6 we need only to prove that f is a bijection.


Figure (Theorem 1.8.7): γYi ∼ γY

(1) f is surjective. Let a ∈ X and b = f(a) ∈ Y. Given y0 ∈ Y. To show theexistence of x0 such that f(x0) = y, we consider the segment from b to y0 in Y:

y(t) := b(1− t) + ty0, 0 ≤ t ≤ 1.

Theorem 1.8.4 =⇒ ∃ open neighborhoods Ua of a in X and Vb of b in Y such thatf : Ua → Vb is a diffeomorphism. Set

x(t) := f−1(y(t)) for y(t) ∈ Vb.

Claim 1: ∃maximum value t0 ∈ [0, 1] such that x(t) is defined for ∀ t ∈ [0, t0].Assume that x(t) is defined on [0, t0), then

x′(t) = [f−1(x(t))]−1y′(t), ∀ 0 ≤ t < t0 ≤ 1.

So||x′(t)||X ≤ M||y′(t)||Y = M||y0 − b||Y, 0 ≤ t < t0 ≤ 1.

By the mean-value theorem, Theorem 1.6.4,

||x(t)− x(τ)||X ≤ M||y0 − b||Y|y− τ|, ∀ 0 ≤ t, τ < t0 ≤ 1.

Hence limt→t0 x(t) = x(t0) exists and f(x(t0)) = y(t0).Claim 2: t0 = 1.Otherwise t0 ∈ (0, 1). Theorem 1.8.4 applied to the neighborhoods of x(t0)

and y(t0) makes it possible to determine x(t) = f−1(y(t)) in a neighborhood ofx(t0) and y(t0), hence for values t > t0.

From Claim 1 and Claim 2 =⇒ x(1) = f−1(y(1)) =⇒ y0 = f(x(1)) = f(x0).(2) f is injective. Assume that ∃ x1 6= x2 such that f(x1) = f(x2) = y. γY

i ∼ γY

consisting only of y =⇒ contradicting the local inverse function theorem.

1.9. Differentiable equations

Let (X, || · ||) be a real Banach space, ϕ : R → X, and ϕ′(t) is identified withϕ′ : R→ X.

1.9.1. First order differential equation. Consider an open subset U ⊂ R× Xand

(1.9.1)dxdt

= f(t, x), x ∈ X, f : U → X continuous.

1.9. DIFFERENTIABLE EQUATIONS 55

A solution of this differential equation is a C1-function ϕ : R ⊃ I → X such that

(1.9.2) (t,ϕ(t)) ∈ U (∀ t ∈ I) and ϕ′(t) = f(t,ϕ(t)) (∀ t ∈ I).

Here I ⊂ R is an open interval.

(1) Example: X = Rn, x = (x1, · · · , xn), f = ( f 1, · · · , f n) =⇒

dxdt

= f (t, x) =⇒ dxi

dt= f i(t, x1, · · · , xn) (1 ≤ i ≤ n).

(2) An equation of order n on X is

(1.9.3)dnxdtn = f

(t, x,

dxdt

, · · · ,dn−1xdtn−1

),

that is equivalent to n equations of first order on X,

dxdt

= x1,dx1

dt= x2, · · · ,

dxn−1

dt= f(t, x, x1, · · · , xn−1),

where (x, x1, · · · , xn−1) ∈ X× · · · × X (n times).

1.9.2. Existence and uniqueness theorems for the Lipschitz case. Let (X, || ·||X) and (Y, || · ||Y) be Banach spaces, U ⊂ X be open. We say f : X → Y isk-Lipschitz in U if

||f(x1)− f(x2)||Y ≤ k||x1 − x2||X , ∀ x1, x2 ∈ U.

Let I ⊂ R be open, U ⊂ X be an open subset in a Banach space (X, || · ||).We say f : I ×U → X is locally Lipschitz if ∀ (t0, x0) ∈ I ×U ∃ neighborhoodN ⊂ I ×U of (t0, x0) and ∃ k > 0 such that

||f(t, x1)− f(t, x2)|| ≤ k||x1 − x2||, ∀ (t, x1), (t, x2) ∈ N.

Equivalently, f is locally Lipschitz if and only if f(t, ·) is k-Lipschitz locally.

Theorem 1.9.1. (Local existence) (X, || · ||) Banach space, U ⊂ X open, I ⊂ R open,f : I ×U → X continuous and locally Lipschitz, (t0, x0) ∈ I ×U =⇒ ∃ a > 0 such that

dxdt

= f(t, x)

has a solution ϕ : [t0 − a, t0 + a]→ X with ϕ(t0) = x0.

PROOF. Take a closed ball B0 of center x0 with radius ε, a closed interval I0 ofcenter t0, such that (t0, x0) ∈ I0 × B0 ⊂ N ≡ N(t0, x0), the neighborhood in thedefinition of locally Lipschitz mapping. Set

sup(t,x)∈I0×B0

|f(t, x)| ≤ M ∈ [0,+∞).

∀ (t, x) ∈ N we have

||f(t, x)|| ≤ ||f(t, x0)||+ k||x− x0||.Define

xn(t) := x0 +∫ t

t0

f(τ, xn−1(τ))dτ, n = 1, 2, · · · , t ∈ I0.


For n = 1,

||x1(t)− x0|| ≤ |t− t0| supt0≤τ≤t

||f(τ, x0)|| ≤ M|t− t0|.

So ∃ a > 0 such that x1(t) ∈ B0 for ∀ t ∈ [t0 − a, t0 + a] (i.e., a < ε/M). In general,if xn−1(t) ∈ B0 (t ∈ [t0 − a, t0 + a])

||xn(t)− x0|| ≤ |t− t0| supt0≤τ≤t

||f(τ, xn−1(τ))|| ≤ M|t− t0| < ε

for ∀ t ∈ [t0 − a, t0 + a]. Moreover

||xn(t)− xn−1(t)|| ≤∫ t

t0

||f(τ, xn−1(τ))− f(τ, xn−2(τ))||dτ

≤ k∫ t

t0

||xn−1(τ)− xn−2(τ)||dτ

so that

||xn(t)− xn−1(t)|| ≤Mkn−1

n!|t− t0|n, t ∈ [t0 − a, t0 + a]

so (xn(t))n≥1 is a Cauchy sequence in B0 for ∀ t ∈ [t0− a, t0 + a] =⇒ limn→∞ xn(t) =:x(t) ∈ B and

x(t) =∫ t

0f(τ, x(τ))dτ + x0, x(0) = x0.

Thus dx(t)/dx = f(t, x(t)).

If f is linear and continuous in x, on X, then it is globally Lipschitz on X:

||f(t, x1)− f(t, x2)|| ≤ k||x1 − x1||, ∀ x1, x2 ∈ X.

In this case, the solution of dx/dt = f(t, x) exists globally and uniquely.

Theorem 1.9.2. (Global uniqueness theorem) Under the same hypotheses as for The-orem 1.9.1 =⇒ ∃ maximum interval J ⊂ I of t0 for which ∃! solution ψ : J → X of theequation

dxdt

= f(t, x), ψ(t0) = x0.

It is called the maximal solution for the initial value (t0, x0).

PROOF. (1) Local uniqueness. Let ϕ1(t) and ϕ2(t) be two solutions with ϕ1(t0) =ϕ2(t0) = x0. If (t, ϕ1(t)) and (t, ϕ2(t)) are in N = N(t0, x0), then

||ϕ1(t)− ϕ2(t)|| ≤ k|t− t0| supt0≤τ≤t

||ϕ1(τ)− ϕ2(τ)||.

When |t− t0| < 1/k, we get ϕ1(t) = ϕ2(t).(2) Global uniqueness. By (1) we see that

ϕ1 and ϕ2 are solutions inj 3 t0 and ϕ1(t0) = ϕ2(t0)

=⇒ ϕ1(t) = ϕ2(t), ∀ t ∈ j.

Consider the set

Σ := (j, ϕ) : ϕ is a solution on j 3 t0 such that ϕ(t0) = x0 .

1.10. PROBLEMS AND REFERENCES 57

∀ (j1, ϕ1) and (j2, ϕ2) in Σ, we must have ϕ1 = ϕ0 on j1 ∩ j2. Define (J, ψ) :=(∪j∈Σ j,∪ϕ∈Σ ϕ).

Example 1.9.3. (1) ∃ solution which does NOT exist in R but f is continuous andlocally Lipschitz on R× X. Consider

dxdt

= x2, X = R, (x0, t0) with x0 > 0 and t0 = 0.

The maximum solution is

ϕ(t) =x0

1− tx0, t ∈ (−∞, 1/x0).

(2) A differential equation depending on a parameter λ ∈ T ⊂ Y (a topologicalspace):

dxdt

= f(t, x; λ).

If||f(t, x; λ)|| ≤ M, ∀ (t, x, λ) ∈ I × B× T

and

||f(t, x1; λ)− f(t, x2; λ)|| ≤ k||x1 − x2||, ∀ t ∈ I, λ ∈ T, x1, x2 ∈ B,

then ϕ(t, λ) is a continuous solution of λ ∈ T and t ∈ [t0 − a, t0 + a].

1.10. Problems and references

Problem 1: Clifford algebra and Spin(4). Let V s,n−s, s ∈ Z≥1 and s ≤ n,be an n-dimensional vector space over R with inner product (v|w) and basis(ei)1≤i≤n such that

(ei|ej) = 0, i 6= j,

(ei|ej) = 1, i = j = 1, · · · , s,

(ei|ej) = −1, i = j = s + 1, · · · , n.

Introduce a product v ·w of vectors in V s,n−s which is associate and distributivewith respect to addition and which satisfies the condition

v ·w + w · v = 2(v|w).

The resulting algebra of all possible sums and products is called the Clifford alge-bra C(V s,n−s) of V s,n−s. Observe that

ei · ej + ej · ei = ±2δij,

(ei)2 := ei · ei = ±1,

v2 = (v|v),ei · ej = −ej · ei, i 6= j.

The Clifford algebra is itself a linear space of dimension ∑0≤p≤n (np) = 2n with

basis1, eI , eI1 · eI2 =: eI1 I2 , · · · , e1 · e2 · · · · · en =: e1···n,

where capital letters label ordered natural numbers: Ij < Ij+1.


(1.1) Show that C(V0,1) is C and that C(V0,2) is the algebra of quaternions.(1.2) The linear subspace of C(V s,n−s) spanned by the (n

p) products (eI1···Ip) isdenoted by Cp(V s,n−s). The linear subspaces

C+(V s,n−s) :=⊕

p evenCp(V s,n−s), C−(V s,n−s) :=

⊕p odd

Cp(V s,n−s)

are called the even and odd subspaces of C(V s,n−s). C+(V s,n−s) is also asubalgebra of V(V s,n−s). The dimension of both C+(V s,n−s) and C−(V s,n−s)is 2n−1. The algebra C+(V s,n−s) is isomorphic to the Clifford algebraC(V s,n−1−s) for certain values of s.

Show that the even subalgebra of the Dirac algebra C(V1,3) is thePauli algebra C(V3,0) and continue the sequence until R are reached.

(1.3) Show that the center Z(V s,n−s) of the algebra C(V s,n−s) is C0(V s,n−s)when n is even and C0(V s,n−s) + Cn(V s,n−s) when n is odd.

Problem 2: Consider the mapping

I : X −→ R, x 7−→∫ 1

0

[1 + |x′(t)|2

]1/4dt.

HereX :=

x ∈ C1([0, 1]) : x(0) = 0, x(1) = 1

.

It is clear that I(x) > 1 for every x ∈ X.(2.1) Under the norm || · ||C1([0,1]), X is also a Banach space and I is continuous.(2.2) Show that I has no upper bound.(2.3) Show that limr→1− I(xr) = 1, where

xr(t) :=

0, t ∈ [0, r],

−1 +[1 + 3(x−r)2

(1−r)2

]1/2, t ∈ [r, 1].

with r ∈ (0, 1). This means that infX I = 1.

Problem 3: Let Ω ⊂ Rn be open, U ⊂ C2(Ω), and V ⊂ C0(Ω). A quasi-linearsecond order operator is

P : U −→ V, u 7−→ Pu := aij(x, u, Du)∂2u

∂xi∂xj + bi(x, u, Du)∂u∂xi .

Assume that aij and bi are C1 functions on Ω× I× I1× · · · × In, where I, I1, · · · , Inare closed intervals in R such that if u ∈ U then u(x) ∈ I, ∂u/∂xi(x) ∈ Ii for allx ∈ Ω.

(3.1) Show that if u0 ∈ U then the mapping P is differential at u0, and computeits differential P′(u0) at u0.

(3.2) If moreover aij and bi are C1,α × C2 × · · · × C2 functions on Ω× I × I1 ×· · ·× In. P′(u0) is still the differential of u 7→ Pu, considered as a mappingfrom C2,α(Ω) into C0,α(Ω).

1.10. PROBLEMS AND REFERENCES 59

References:[1] Choquet-Bruhat, Yvonne; DeWitt-Morette, C.; Dillard-Bleick, M. Analysis,

manifolds and physics, Part I: Basics, Revised Edition, Elsevier, 2010.

[2] Kothe, G. Topological vector spaces, Second printing, Revised, Grundlehrender mathematischen Wissenschaften, 159, Springer-Verlag, 1983.

[3] Morrey, Charles, B. Multiple integrals in the calculus of variations, Reprint ofthe 1966 Edition, Classics in Mathematics, Springer, 2008.

CHAPTER 2

Differentiable manifolds

2.1. Differentiable manifolds

2.1.1. Definitions. A (topological) manifold is a Hausdorff topological spaceX such that ∀ x ∈ X has a neighborhood homeomorphic to Rn. The dimensionof X is n.

(1) A chart (U ,ϕ) of a manifold X is an open set X of X , called the domainof the chart, together with a homeomorphism ϕ : U → U := ϕ(U) ⊂ Rn

(open).(2) Given an arbitrary chart (U ,ϕ) of X . ∀ x ∈ U ⊂ X , we have ϕ(x) ∈ Rn.

The coordinates (x1, · · · , xn) of ϕ(x) are called the coordinates of x in thechart (U ,ϕ). A chart (U ,ϕ) is also called a local coordinate system.

(3) An atlas of class Ck on a manifold X is a set ((Uα,ϕα))α∈I of charts of Xsuch that X = ∪α∈IUα and (ϕα)α∈I satisfies the compatibility condition:

ϕβ ϕ−1α : ϕα(Uα ∩ Uβ) −→ ϕβ(Uα ∩ Uβ), (x1, · · · , xn) 7−→ (y1, · · · , yn),

with yj = yj(x1, · · · , xn) being of Ck.(3.1) Two Ck-atlases ((Uα,ϕα))α∈I and ((Uα′ ,ϕα′))α′∈I′ are equivalent (use

the symbol ∼) if ϕα′ ϕ−1α : ϕα(Uα ∩ Uα′) → ϕα′(Uα ∩ Uα′) are also

Ck.(3.2) A Ck-structure on X is an equivalence class of Ck-atlas.

♣ Exercise: Show that ∼ is indeed an equivalence relation.

(4) A Ck-manifold X is a topological manifold X together with a Ck-structureon X. When k = ∞, we say C∞-manifold as smooth manifolds.

(5) A real analytic/ Cω manifold is defined similarly, except that we nowrequire that the mappings ϕβ ϕ−1

α have to be real analytic.(6) A complex analytic/complex manifold is defined similarly with Cn re-

placing Rn and the mappings ϕβ ϕ−1α being real analytic or holomor-

phic.

♣ Exercise: (i) The double cone (x, y, z) ∈ R3 : x2 − y2 − z2 = 0 is NOTa manifold under the topology included by the usual topology on R3. However(x, y, z) ∈ R3 : x2 − y2 − z2 = 0 and x ≥ 0 is a C0-manifold that is homeomor-phic to R2.

61

62 2. DIFFERENTIABLE MANIFOLDS

Figure 2.1: f is differentiable at x

♣ Exercise: (ii) We require Hausdorff condition in the definition of mani-folds. The following topological space

X :=((−∞, 01]× 1

)∪((−∞, 02]×−1

)∪((0,+∞)× 0

), 01 = 02 = 0,

is NOT Hausdorff. Here the topology on X is generated by a basis of open neigh-borhoods of the form (−ai, 0i] ∪ (0, b) (i = 1, 2).

We have NOT included here in the definition of a differentiable manifold anaxiom of countability of the domains of the charts.

Let X be a Ck-manifold, with k ∈ 0, 1, · · · , ∞, ω.(1) Let (U ,ϕ) be a chart at x (i.e., x ∈ U ). Then (see Figure 2.1), any function

f : X → R gives a map f ϕ−1 : Rn ⊃ ϕ(U )→ R.(2) f is differentiable at x if f ϕ−1 is differentiable at ϕ(x).

(2.1) The definition does NOT depend on the chart: f ϕ−1 = ( f ϕ−1) (ϕ ϕ−1) ∈ C0 Ck = Ck.

(2.2) f is Cr-differentiable at k if f ϕ−1 is Cr-differentiable at ϕ(x). Herek ≥ r.

(2.3) f : X → R is Cr on X if it is Cr at ∀ x ∈ X . Let

(2.1.1) Cr(X ) := all Cr-functions f : X → R .

2.1. DIFFERENTIABLE MANIFOLDS 63

Let X be a Ck-manifold.(1) The i-th coordinate function ai on Rn:

(2.1.2) ai : Rn −→ R, (u1, · · · , un) 7−→ ui.

(2) ∀ chart (U ,ϕ) of X let

(2.1.3) ϕi := ai ϕ : U −→ R, x 7−→ ai(ϕ(x)) = xi.

Then

(2.1.4) ai := ϕi ϕ−1 : ϕ(U ) −→ R, (x1, · · · , xn) 7−→ xi.

Direct product manifolds:

(X , TX , ((Uα,ϕα))α∈I)×(Y , TY , ((Uβ,ϕβ))β∈J

):=

(X ×Y , TX×Y , ((Uα ×Uβ,ϕα ×ϕβ))(α,β)∈I×J

),

whereϕα ×ϕβ(x, y) := (ϕα(x),ϕβ(y)), ∀ x ∈ Uα, y ∈ Uβ.

Then

(2.1.5) dim(X ×Y) = dimX + dimY .

2.1.2. Diffeomorphisms. Let X and Y be Ck-manifolds with dim X = m anddimY = n. Consider a mapping f : X → Y .

(1) (U ,ϕ) chart of X , (V , ψ) chart of Y =⇒

(2.1.6) ψ f ϕ−1 : ϕ(U ) −→ ψ(V).(2) f is Cr-differentiable at k (k ≥ r) if ψ f ϕ−1 is Cr-differentiable at ϕ(x):

ψ f ϕ−1 : (x1, · · · , xm) 3 ϕ(U ) −→ (y1, · · · , yn) ∈ ψ(V), yα = f α(x1, · · · , xm).

(3) f is a Cr-mapping if f is Cr at x ∈ X .(4) f is a Cr-diffeomorphism if f is a bijection and f, f−1 are Cr. In particular,

f is a diffeomorphism if f is a C1-diffeomorphism.(5) For g : Y → R, define the pull back under f of g by

(2.1.7) f∗g := g f.

(6) Let

C∞M := C∞-manifolds, VectR := vector spaces over R

andHomC∞

M(X ,Y) := C∞-mappings X → Y.

Define

C : C∞M −→ VectR,

X 7−→ C∞(X ),f : X → Y 7−→ f∗ : C∞(Y)→ C∞(X ).


Figure 2.2: Cr-differentiable mapping.

♣ Exercise: Show that C is a contravariant functor.

2.1.3. Lie groups. A Lie group G is a group that is also a smooth manifoldsuch that the smooth structure is compatible with the group structure,

(2.1.8) G × G −→ G, (x, y) 7−→ xy−1,

is a C∞-mapping.

(1) Two Lie groups G1 and G2 are isomorphic if ∃ diffeomorphism f betweenG1 and G2 which preserves the group structures (so is a group homomor-phism).

(2) G Lie group =⇒ G → G, x 7→ x−1, is C∞.(3) (Rn,+) is a Lie group.(4) GL(n, R) is a Lie group.(5) A one-dimensional Lie group is usually called a one-parameter group.

Theorem 2.1.1. A connected one-parameter group G is isomorphic to R or T := R/Z.

(6) A connected topological group which possesses a neighborhood of theorigin homeomorphic to R is isomorphic to R or T.

2.2. VECTOR FIELDS AND TENSOR FIELDS 65

(7) A local Lie group is a neighborhood of the origin of a Lie group. A localone parameter group is isomorphic to an interval I ⊂ R containing theorigin.

2.2. Vector fields and tensor fields

2.2.1. Tangent vectors. Let X be a smooth manifold of dimension n. The tan-gent vector space at x ∈ X is a vector space TxX that is isomorphic to Rn.

(1) Definition A. Define

(2.2.1) C∞(x) := functions defined and C∞ on some neighborhood of x .

Any element of C∞(x) is a pair ( f ,U f ) with U f ⊂⊂ X being a neighbor-hood of x.

(1.1) A tangent vector vx : C∞(x) → R is a linear mapping satisfying theLeibniz rule

vx(α f + βg) = αvx( f ) + βvx(g), α, β ∈ R,vx( f g) = f (x)vx(g) + g(x)vx( f ), f , g ∈ C∞(x).

Consequently,

vx(1) = vx(1 · 1) = 2vx(1) =⇒ vx(1) = 0

and

vx(α) = vx(α · 1 + 0 · 1) = αvx(1) = 0, ∀ α ∈ R.

(1.2) vx is also called a derivation and vx( f ) is the directional derivativeof f along vx.

(1.3) Set

(2.2.2) TxX := tangent vectors vx to X at x.

Under the operation

(αvx + βux)( f ) := αvx( f ) + βux( f ),

TxX is a vector space.(1.4) We say that f , g ∈ C∞(x) have the same germ at x, written as f ∼ g,

if ∃ W ⊂ U f ∩ Ug such that f ≡ g onW . Set

(2.2.3) [ f ] := g ∈ C∞(x) : g ∼ f

the germ of f , and

(2.2.4) Fx := [ f ] : f ∈ C∞(x) = C∞(x)/ ∼,

the germs at x. Define + and · to be

[ f ] + [g] ≡ [( f ,U f )] + [(g,Ug)] := [( f + g,U f ∩ Ug)],

[ f ] · [g] ≡ [( f ,U f )] · [(g,Ug)] := [( f · g,U f ∩ Ug)].

(1.5) A tangent vector then is a derivation on Fx.(1.6) In the chart (U ,ϕ), the components of vx ∈ TxX are

(2.2.5) vi := vx(ϕi) with ϕi = ai ϕ.


(1.7) For f ∈ C∞(x0) and a chart (U ,ϕ) of x0, applying the mean valuetheorem to f ϕ−1, we obtain

f (x) = f (x0) + [ϕi(x)− ϕi(x0)]∂( f ϕ−1)

∂xi

∣∣∣∣ϕ(x0)+[ϕ(x)−ϕ(x0)]s

for some s ∈ (0, 1). Then ∀ vx0 ∈ Tx0X

(2.2.6) vx0( f ) = vi ∂( f ϕ−1)

∂xi

∣∣∣∣ϕ(x0)

, vi := vx0(ϕi).

(1.8) Set

(2.2.7)∂

∂xi

∣∣∣∣x0

f :=∂( f ϕ−1)

∂xi

∣∣∣∣ϕ(x0)

.

Then ∀ vx0 ∈ Tx0X

(2.2.8) vx0 = vi ∂

∂xi

∣∣∣∣x0

and (∂/∂xi|x0)1≤i≤n forms a natural basis for Tx0X . Consequently

(2.2.9) dim Tx0X = dimX = n

and

(2.2.10) Tx0X −→ Rn, vx0 = vi ∂

∂xi

∣∣∣∣x0

7−→ (v1, · · · , vn) =: v.

(1.9) If f has a critical point at x0 (i.e., ∂( f ϕ−1)/∂xi|ϕ(x0)= 0), then

vx0( f ) = 0, and conversely,

vx0( f ) = 0 (∀ vx0) ⇐⇒ ∂( f ϕ−1)

∂xi

∣∣∣∣ϕ(x0)

= 0.

(2) Definition B. Let (U ,ϕ) and (U ,ϕ′) be two charts (see Figure 2.3). Write

f := f ϕ−1, f ′ := f ′ ϕ′−1, f = f ′ ϕ′ ϕ−1.

Then

∂ f (x1, · · · , xn)

∂xi =∂ f ′(x′1, · · · , x′n)

∂x′ j∂(aj ϕ′ ϕ−1)(x1, · · · , xn)

∂xi

or∂ f∂xi =

∂ f ′

∂x′ j∂x′ j

∂xi .

This gives for ∀ vx ∈ TxX

vx( f ) = vi ∂ f∂xi

∣∣∣∣ϕ(x)

= vi ∂ f ′

∂x′ j

∣∣∣∣ϕ′(x)

∂x′ j

∂xi

∣∣∣∣ϕ(x)

and then

v′ j = vi ∂x′ j

∂xi

∣∣∣∣ϕ(x)

for each j = 1, · · · , n.


(2.1) The set (vi)1≤i≤n determines a vector v ∈ Rn and the set (v′ i)1≤i≤ndetermines another vector v′ ∈ Rn such that

(2.2.11) v′ = D(ϕ′ ϕ−1)

∣∣∣∣ϕ(x)

v.

(2.2) A tangent vector vx is an equivalence class [(U ,ϕ, v)] with (U ,ϕ)being a chart and v ∈ Rn, where (U ,ϕ, v) is equivalent to (U ,ϕ′, v′)if and only if (2.2.11) is true.

(3) Definition C. The last equivalent definition comes from differentiable curves.(3.1) A (parametrized) curve γ on X is a mapping from I ⊂ R into X by

I −→ X , t 7−→ γ(t).

(3.2) A differentiable curve γ at x0 is a differentiable mapping from I ⊂R into X such that 0 ∈ I and γ(0) = x0. Such differentiable curvesforms a set C1

x0(0).

(3.3) ∀ γ ∈ C1x0(0) and ∀ f ∈ C∞(x0) =⇒ f γ ∈ C1(0) and f γ(0) =

f (x0). The tangent vector to γ at x0 is a mapping1

(2.2.12) vγx0 : Fx0 −→ R, [ f ] 7−→ d

dt

∣∣∣∣t=0

( f γ)(0) =: vγx0( f ).

(3.4) γ1, γ2 ∈ C1x0(0) are said to be tangent at x0 if

vγ1x0 ( f ) = vγ2

x0 ( f ), ∀ f ∈ Fx0 .

(3.5) LetTx0X :=

[γ] : γ ∈ C1

x0(0)

.

(3.6) Given a tangent vector vx0 ∃ γ ∈ C1x0(0) (by Theorem 1.9.1) such that

vγx0 = vx0 .

(3.7) If vx0 = vγx0 , then (see Figure 2.4)

(vγx0)

i = vγx0(ϕi) =

ddt

∣∣∣∣t=0

(ϕi γ)(t) =dγi(t)

dt

∣∣∣∣t=0

and

(2.2.13) vγx0( f ) =

ddt

∣∣∣∣t=0

( f ϕ−1 ϕ γ) =∂( f ϕ−1)

∂xi

∣∣∣∣ϕ(x0)

dγi

dt

∣∣∣∣t=0

.

♣ Exercise: Show that (Fx,+, ·) is an algebra.

Let f : X → Y be differential at x0, dimX = n and dimY = p.(1) Define a linear mapping

(2.2.14) Txf ≡ f∗,x ≡ dxf ≡ f′(x) : TxX −→ Tf(x)Y , v 7−→ w

by

(2.2.15) w(h) := v(h f), ∀ h ∈ C∞(y)

where y := f(x).

1Since vγx0 ([ f ]) is independent of the choice of the representative f we can define vγ

x0 ( f ).


Figure 2.3: Definition B

Figure 2.4: Definition C


(2) Let

C∞M,∗ = category of

objects : pointed smooth manifold (X , x),

morphisms :HomC∞

M,∗((X , x), (Y , y)) =

f : X → Y differential and f(x) = yDefine a covariant functor

T∗ : C∞M,∗ −→ VectR

(X , x) 7−→ TxX ,f : (X , x)→ (Y , y) 7−→ T∗f : TxX −→ TyY .

♣ Exercise: Show that C∞M,∗ is a category and T∗ is a covariant functor.

Proposition 2.2.1. Let f : X → Y be differential at x and choose local coordinates(U ,ϕ, xi) on X and (V , ψ, yα) on Y =⇒ If v ∈ Rn and w ∈ Rp represent respectivelyv ∈ TxX and w ∈ TyY such that w = Txf(v), then

w = f′(x1, · · · , xn)v (⇐⇒ w = f′(x)v),

where f := ψ f ϕ−1 : ϕ(U )→ ψ(V).

PROOF. Compute, where yα := fα(x1, · · · , xn),

w(h) = (Txf(v)) (h) = v(h f) = vi ∂(h f ϕ−1)

∂xi

= vi ∂(h ψ−1 f)∂xi = vi ∂(h ψ−1)

∂yα

∂fα

∂xi = wα ∂(h ψ−1)

∂yα.

Thus wα = vi(∂fα/∂xi).

Examples:(1) Given a curve γ : R → X , a vector vγ

x tangent to γ at x := γ(0), anda mapping f : X → Y =⇒ Txf(vγ

x ) is tangent to f γ at y := f(x) =f γ(0).

PROOF. ∀ g ∈ C∞(y) =⇒(Txf(vγ

x ))(g) = vγ

x (g f) =ddt

∣∣∣∣t=0

(g f γ)(t) =ddt

∣∣∣∣t=0

(g (f γ))(t).

(2) Under hypothesis in (1) =⇒

T∗f(vγx ) =

ddt

∣∣∣∣t=0

(f γ)(t).

(3) In spherical coordinates (r, θ, ϕ) in R3, ∂/∂r is the unit tangent vector tothe curve: θ = constant, ϕ = constant =⇒ ??? components of ∂/∂r in(x1, x2, x3).


Figure 2.5: Spherical coordinates

PROOF. Note that

ϕ1(x) = (r, θ, ϕ), ϕ2(x) = (x1, x2, x3), ai ϕ2(x) = xi

and ϕ2 ϕ−11 : (r, θ, ϕ) 7→ (x1, x2, x3) is given by

x1 = a1 ϕ2 ϕ−11 (r, θ, ϕ) = r sin θ cos ϕ,

x2 = a2 ϕ2 ϕ−11 (r, θ, ϕ) = r sin θ sin ϕ,

x3 = a3 ϕ2 ϕ−11 (r, θ, ϕ) = r cos θ.

Then(T(ϕ2 ϕ−1

1 )∂

∂r

)(h) =

∂

∂r(h ϕ2 ϕ−1

1 ) =∂h∂xi

∂(ai ϕ2 ϕ−11 )

∂r.

Hence∂

∂r∼= T(ϕ2 ϕ−1)

∂

∂r= sin θ cos ϕ

∂

∂x1 + sin θ sin ϕ∂

∂x2 + cos θ∂

∂x3 .

2.2.2. Fibre bundles. A bundle (E ,B, π,F ) consists of(i) topological spaces E and B,

(ii) continuous surjective mapping π : E → B such that ∀ x ∈ B the fiber atx, π−1(x) =: Ex, is homeomorphic to a topological space F .

We call B the base and F the typical fibre.


(1) A fibre bundle (E ,B, π,F ,G) is a bundle (E ,B, π,F ) together with atopological group G (called a a structure group) of homeomorphisms ofF onto itself, and an open covering (Ui)i∈I of B, such that

(1.1) locally the bundle is a trivial bundle:

π−1(Ui)ϕi∼=//

π

Ui ×F

yysssssssssss

Ui

p //

π

(π(p),ϕ•i (p))

xxqqqqqqqqqq

π(p)

We call ((Ui,ϕi))i∈I a family of local trivializations of the bundle.For ∀ x ∈ Ui, ϕ•i,x := ϕ•i |Fx : Fx → F is homeomorphic.

(1.2) ∃ correlation of the trivial subbundles defined on the open covering(Ui)i∈I of B. ∀ x ∈ Ui ∩ Uj we have

ϕ•j,x ϕ•i,x ∈ G

(1.3) The induced mapping gij : Ui ∩ Uj → G given by

(2.2.16) gij(x) := ϕ•j,x ϕ•i,x, ∀ x ∈ Ui ∩ Uj

called the transition functions (see Figure 2.6), are continuous. Notethat

(2.2.17) gij(x)g jk(x) = gik(x), ∀ x ∈ Ui ∩ Uj ∩ Uk.

(2) A vector bundle is a fibre bundle (E ,B, π,F ,G) where F is a vectorspace and G is the linear group.

(3) A bundle morphism between two bundles (E1,B1, π1,F1) and (E2,B2, π2,F2)

is a pair of mappings (f, f) such that the diagram

E1f−−−−→ E2

π1

y yπ2

B1 −−−−→f

B2

is commutative, and f : E1,x → E2,f(x2)is linear for each x ∈ B1.

(4) A bundle category Bundle consists of bundles and bundle morphisms.(5) A fibre bundle (E ,B, π,F ,G) is said to be a Ck fibre bundle if E ,B,F are

Ck manifolds, π is a Ck mappings, G is a Lie group, and the covering ofB being the domains of an admissible atlas, the mappings gij are Ck.

(6) A chart (U ,ϕ) on E defines fibre coordinates on E if the mapping ϕ :U → Rn+p is a bundle morphism, with Rn+p having the natural bundlestructure Rn+p = Rn × Rp. Here (E ,B, π,F ,G) is a differential fibrebundle, dim E = n + p, dimB = n.

♣ Exercise: Show that Bundle is a category.

Tangent bundle. Let X be a differentiable manifold of dimension n. Define

(2.2.18) TX := (x, vx) : x ∈ X , vx ∈ TxX .


Figure 2.6: Transition functions

Then(TX ,X , π, Rn, GL(n, R))

is a fibre bundle.

(1) Fibre at x: TxX ; Typical fibre: Rn; Projection: π : TX → X , (x, vx) 7→ x.(2) Covering of X :

X =⋃i∈IUi, where (Ui, ψi)i∈I is an atlas of X .

(3) Homomorphism ϕi:

π−1(Ui)ϕi //

π

Ui ×Rn

yysssssssssss

Ui

withϕi := (π, ψ′i π2),

where π2(x, vx) = vx ∈ TxX and ψ′i(vx) = v is the representative of vxin the chart (Ui, ψi). The fibre coordinates on TX are given by

(ψi, 1) (π, ψ′i π2) : π−1(Ui) −→ Rn ×Rp = Rn+p,

p = (x, vx) 7−→ (x1, · · · , xn, v1, · · · , vn).(2.2.19)


(4) Structure group:

GL(n, R) = n× n real matrices A with det(A) 6= 0.

∀ x ∈ Ui ∩ Uj, we have

ψ′i,x : TxX → Rn, vx 7→ v and ψ′j,x : TxX → Rn, vx 7→ w

so that ψ′j,x ψ′−1i,x : Rn → Rn, v 7→ w.

(5) X is Ck =⇒ TX is Ck−1.Frame bundle. Let X be a C∞-manifold.

(1) A frame ρx in TxX is a set of n linearly independent vectors (e1, · · · , en)which can be expressed as a linear combination of the elements of a par-ticular basis (e1 , · · · , en) of TxX :

ei = ajiej , (aj

i) ∈ GL(n, R).

Thenframes in TxX ←→ GL(n, R).

Let

(2.2.20) FX := (x, ρx) : x ∈ X and ρx is a frame at x.

Then(FX ,X , π, GL(n, R), GL(n, R))

is a fibre bundle.(2) A frame ρx can be thought of as a nonsingular linear mapping

ρx : Rn −→ TxX , (v1, · · · , vn) 7−→ vx := ∑1≤i≤n

viei

if ρx = (e1, · · · , en).

A fibre bundle (E ,B, π,F ,G) in which F and G are isomorphic and in whichG acts on F by left translation, is called a principal fibre bundle. For example,(FX ,X , π, GL(n, R), GL(n, R)) is a principal fibre bundle.

(1) Let (E ,B, π,F ,G) be a principal fibre bundle. Let (Ui)i∈I be the opencovering of X used to define the fibre bundle structure. Let p ∈ Ex andx ∈ Ui, define

(2.2.21) gi := ϕ•i,x(p)

where we identify F with G:

Exϕ•i,x−−−−→∼=

F −−−−→∼= G

and ∀ g ∈ G,

(2.2.22) Rg p ≡(

Rg p)

i:= ϕ•−1

i,x (Rggi) = ϕ•−1i,x (gig), p ∈ π−1(Ui) =

⋃x∈Ui

Ex.


Figure 2.8: (Rg p)i = (Rg p)j

Since Rg1 Rg2 p = Rg2g1 p, it follows that (Rg)g∈G is a group anti-isomorphicto G, acting on the right on π−1(Ui).

(Rg)g∈G ×π−1(Ui) −→ π−1(Ui),

(Rg, p ∈ Ex) 7−→ Rg p ∈ Ex

so (Rg)g∈G acts transitively on each fibre Ex, x ∈ π−1(Ui).(2) ∀ p ∈ π−1(Ui ∩ Uj) =⇒ ( see Figure 2.8)(

Rg p)

i=(

Rg p)

j.

PROOF. For p ∈ Ex with x ∈ Ui ∩ Uj =⇒

ϕ•i,x(p) = gi, ϕ•j,x(p) = gj, gi = ϕ•i,x ϕ•−1j,x (gj).

Then gi = gij(x)gj and(Rg p

)j= ϕ−1

j,x (gjg) = ϕ•−1i,x ϕi,x ϕ•−1

j,x (gjg)

= ϕ•−1i,x

(gij(x)gjg

)= ϕ•−1

i,x (gig) =(

Rg p)

i.

(3) Since the mapping Rg is independent of the choice of the open set Uicontaining π(p), it is well-defined over all of E , and we can write

(2.2.23) Rg(p) = ϕ•−1i,x Rg ϕ•i,x(p), x = π(p).


Hence we get the right action of G on E :

(2.2.24) E × G −→ E , (p, g) 7−→ pg := Rg(p).

Since p ∈ Ex =⇒ Rg(p) ∈ Ex, (2.2.24) is a fibre-preserving global right-action.

(4) Rg geometric meaning: connection. Rg : G → G ∼= F ∼= Ex =⇒TeRg : g = TeG → TpEx with π(p) = x.

(5) Examples:(5.1) In general, the left action of G on the principal fibre bundle (E ,X , π,F ,G)

does not define a fibre-preserving global action. Indeed, we, as thesame construction, obtain

gi = ϕ•i,x(p),(

Lg p)

i= ϕ•−1

i,x (Lggi) = ϕ•−1i,x (ggi).

Using ϕ•i,x ϕ•−1j,x (g) = gij(x)g yields gi = gij(x)gj and(

Lg p)

j= ϕ•−1

j,x (ggj) = ϕ•−1i,x ϕ•i,x ϕ•−1

j,x (ggj) = ϕ•−1i,x

(gij(x)ggj

)6=(

Lg p)

i.

(5.2) G = GL(n, R) E = FX . Indeed, ∀ x ∈ Ui and ∀ p ∈ π−1(Ui) andlet ϕ•i,x(p) = gi = (a(i)

µλ) ∈ G ∼= F ∼= Ex. The action of G on the

typical fibre Ex is

Aµα a(j)

αλ = a(i)

µλ ⇐⇒ ϕ•i,x ϕ•−1

j,x = (Aµα)

and the right action of G on E is(Rg p

)i

= ϕ•−1i,x

((a(i)

µα Gα

λ

))= ϕ•−1

i,x

((Aµ

βa(j)βα Gα

λ

))= ϕ•−1

i,x ϕ•i,x ϕ•−1j,x

((a(j)

βα Gα

λ

))= ϕ•−1

j,x

((a(j)

βα Gα

λ

))with g = (Gα

β).

Given a fibre bundle (E ,B, π,F ,G), it it admits an equivalent structure de-fined with a subgroup G1 of G, then ∃ a family of local trivializations with transi-tion functions gij taking their value in G1. We say that G is reducible to G1.

(1) The structure group GL(n, R) of TRn is reducible to 1. In general, ∀fibre bundle with base Rn is reducible to a trivial bundle (since Rn is acontractible space).

(2) A principal fibre bundle (E ,X , π,F ,G) is said to be reducible to the prin-cipal fibre bundle (E1,X , π1,F1,G1) if

(2.1) G1 < G and E1 ⊂ E ,(2.2) the injection f : E1 → E is a bundle morphism commuting with the

action of G1:

G1 × E1Rg−−−−→ E1

1×fy yf

G1 × E −−−−→Rg

E


with

π(f(p)) = π1(p), ∀ p ∈ E1,

f(Rg(p)) = Rg(f(p)), ∀ p ∈ E1, ∀ g ∈ G1.

(3) FX is reducible to the bundle of orthogonal frames or Lorentz frames.

Theorem 2.2.2. A smooth principal fibre bundle (E ,X , π,F ,G) is reducible to(E1,X , π1,F1,G1) with G1 a Lie subgroup of the Lie group G ⇐⇒ ∃ family of localtrivializations whose transition functions take their value in G1.

2.2.3. Vector fields. Given a bundle (E ,B, π,F ).(1) A cross-section is a mapping f : B → E such that π f = 1B .

Theorem 2.2.3. A principal fibre bundle (E ,X , π,F ,G) is trivial⇐⇒ ∃ a continuouscross-section.

PROOF. (i) triviality =⇒ cross-section. Define

α : X −→ X × G, x 7−→ (x, f(x))

Then f is a continuous cross-section.(ii) cross-section =⇒ triviality. Let f : X → E be a continuous cross-section.

Given p ∈ E , ∃ x ∈ X such that p ∈ Ex, ∃! g0 ∈ G such that

p = Rg0(f(x)).

Define

α : E −→ X × G, p 7−→ (x, g0)

which preserves the group structure of the fibres:

α(Rg′ p) = α(

Rg0g′ f(x))= (x, g0g′) = (x, g0)g′ = Rg′α(p)

for all g′ ∈ G and p ∈ E . Note that α(f(x)) = (x, e), e ∈ G identity.

Let X be a Cr-manifold of dimension n, where r ∈ 1, 2, · · · ,+∞, ω.(1) A vector field v on X is a cross-section of (TX ,X , π, Rn).

(1.1) ∀ x ∈ X , the vector field v associates a tangent vector vx ∈ TxX .Thus

(2.2.25) v : X −→ TX , x 7−→ (x, vx) or vx.

(2) A Cr-vector field is a vector field v on X such that v : X → TX is Cr

(r ≤ k− 1).(2.1) A vector field v is Cr ⇐⇒ ∀ chart of an admissible atlas on X the n

functions vi are of class Cr.


(2.2) A vector field v can be defined as a derivation on Ck(X ):

(2.2.26) v : Ck(X ) −→ Ck−1(X ), f 7−→ v( f )

with vx( f ) = (v( f ))(x). Locally

(2.2.27) (v( f ))(x) = vx( f ) = vix

∂ f∂xi

∣∣∣∣x

=⇒ v f = vi ∂ f∂xi .

(3) Lie algebra X(X ) (where X is a smooth manifold of dimension n):

(2.2.28) X(X ) := C∞ vector fields on X.

Addition in X(X ):

(v + w)( f ) := v( f ) + w( f ), ∀ v, w ∈ X(X ), ∀ f ∈ C∞(X ).

Multiplication of v ∈ X(X ) by g ∈ C∞(X ):

C∞(X )×X(X ) −→ X(X ), (g, v) 7−→ gv, (gv)( f ) := g(v( f )).

Then X(X ) is a module on the ring C∞(X ). However, X(X ) is not closedunder multiplication defined by (vw)( f ) := v(w( f )):

(vw)( f g) = v(w( f g)) = v ( f (wg) + g(w f ))= f ((vw)g) + g ((vw) f ) + [(v f )(wg) + (vg)(w f )] .

Lie bracket on X(X ):

(2.2.29) [·, ·] : X(X ×X(X ) −→ X(X ), (v, w) 7−→ [v, w] := vw− wv =: Lvw.

Here Lvw is called the Lie derivative of w in the direction of v. Since

(vw) f = vi ∂

∂xi

(wj ∂ f

∂xj

)= viwj ∂2 f

∂xi∂xj + vi ∂wj

∂xi∂ f∂xj ,

we get

(2.2.30) [v, w] f =

(vi ∂wj

∂xi − wi ∂vj

∂xi

)∂ f∂xj ∈ X(X ).

Observe that: [·, ·] is distributive with respect to addition and anti-commutative,but is not associative. We have the Jacobi identity:

(2.2.31) [v1, [v2, v3]] + [v2, [v3, v1]] + [v3, [v1, v2]] = 0.

♣ Exercise: (i) Show that X(X ) is a module on the ring C∞(X ). (ii) Verify(2.2.31).

(4) A moving frame (if n = 4, vierbein or tetrad) is a set of n linearly in-dependent C∞ vector fields (ei)1≤i≤n which form a basis for the moduleX(U ), U ⊂ X .

(5) The image of a vector at x ∈ X under a differentiable map f : X → Y isgiven by

(f′vx)(g) := vx(g f), vx ∈ TxX , g ∈ C∞(x).

Then f′vx ∈ Tf(x)Y .


(5.1) When f is invertible, we get, for x = f−1(y) and v ∈ X(X ),[(f′v)(g)

](y) = [v(g f)](x) = [v(g f)](f−1(y))

so that(f′v)(g) = [v(g f)] f−1.

(5.2) v a Cr-vector field and f : X → Y a Cr+1-diffeomorphism =⇒ f′v isa Cr-vector field on Y .

(5.3) We say that a vector field v on X and a vector field w on Y are f-related if w = f′v.

2.3. Lie groups

2.4. Exterior differential forms

2.5. Integration

2.6. Exercises and problems

Documents

Analysis on manifolds · 1.5. Linear functional analysis 36 1.6. Differentiable calculus on Banach spaces 41 1.7. Calculus of variations 48 1.8. Implicit function theorem and inverse