Schneider-Nonarchimedean Functional Analysis 2

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Nonarchimedean Functional Analysis

Peter Schneider

Version: 25.10.2005

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This book grew out of a course which I gave during the winter term 1997/98at the Universitat Munster. The course covered the material which here ispresented in the first three chapters. The fourth more advanced chapter wasadded to give the reader a rather complete tour through all the important aspectsof the theory of locally convex vector spaces over nonarchimedean fields. Thereis one serious restriction, though, which seemed inevitable to me in the interestof a clear presentation. In its deeper aspects the theory depends very much onthe field being spherically complete or not. To give a drastic example, if the fieldis not spherically complete then there exist nonzero locally convex vector spaceswhich do not have a single nonzero continuous linear form. Although much

progress has been made to overcome this problem a really nice and completetheory which to a large extent is analogous to classical functional analysis canonly exist over spherically complete fields. I therefore allowed myself to restrictto this case whenever a conceptual clarity resulted.

Although I hope that this text will also be useful to the experts as a referencemy own motivation for giving that course and writing this book was different.I had the reader in mind who wants to use locally convex vector spaces in theapplications and needs a text to quickly grasp this theory. There are severalareas, mostly in number theory like p-adic modular forms and deformations ofGalois representations and in representation theory of p-adic reductive groups,

in which one can observe an increasing interest in methods from nonarchimedeanfunctional analysis. By the way, discretely valued fields like p-adic number fieldsas they occur in these applications are spherically complete.

This is a textbook which is self-contained in the sense that it requires only somebasic knowledge in linear algebra and point set topology. Everything presentedis well known, nothing is new or original. Some of the material in the last chapterappears in print for the first time, though. In the references I have listed all thesources I have drawn upon. At the same time this list shows to the reader whothe protagonists are in this area of mathematics. I certainly do not belong tothis group.

Munster, May 2001 Peter Schneider

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List of contents

Chap. I: Foundations

1 Nonarchimedean fields

2 Seminorms3 Normed vector spaces

4 Locally convex vector spaces

5 Constructions and examples

6 Spaces of continuous linear maps

7 Completeness

8 Frechet spaces

9 The dual space

Chap. II: The structure of Banach spaces

10 Structure theorems

11 Non-reflexivity

Chap. III: Duality theory

12 c-compact and compactoid submodules

13 Polarity

14 Admissible topologies

15 Reflexivity

16 Compact limits

Chap. IV: Nuclear maps and spaces

17 Topological tensor products

18 Completely continuous maps

19 Nuclear spaces

20 Nuclear maps

21 Traces

22 Fredholm theory

References

Index, Notations

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Chap. I: Foundations

In this chapter we introduce the basic notions and constructions of nonar-chimedean functional analysis. We begin in 1 with a brief but self-contained

review of nonarchimedean fields. The main objective of functional analysis is theinvestigation of a certain class of topological vector spaces over a fixed nonar-chimedean field K. This is the class of locally convex vector spaces. The moretraditional analytic point of view characterizes locally convex topologies as thosevector space topologies which can be defined by a family of (nonarchimedean)seminorms. But the presence of the ring of integers o inside the field K allows foran equivalent algebraic point of view. A locally convex topology on a K-vectorspace V is a vector space topology defined by a class of o-submodules ofV whichare required to generate V as a vector space. In 2 and 4 we thoroughly discussthese two concepts and their equivalence. Throughout the book we usually willpresent the theory from both angles. But sometimes there will be a certain bias

towards the algebraic point of view.The most basic methods to actually construct locally convex vector spaces alongwith concrete examples are treated in 3 and 5. In 6 we explain how thenotion of a bounded subset leads to a systematic method to equip the vectorspace of continuous linear maps between two given locally convex vector spaceswith a natural class of locally convex topologies. The two most important onesamong them are the weak and the strong topology. The important concepts ofcompleteness and quasi-completeness are discussed in 7. The construction ofthe completion of a locally convex vector space is one of the places where we findan algebraic treatment preferable since conceptually simpler. Banach spaces as

already introduced in 3 are complete. They are included in the very importantclass of Frechet spaces. These are the complete locally convex vector spaceswhose topology is metrizable. Their importance partly derives from the validityof the closed graph and open mapping theorems for linear maps between Frechetspaces. These basic results are established in 8 using Baire category theory. Inthe final 9 of this chapter we begin the investigation of the continuous lineardual of a locally convex vector space. Provided the field Kis spherically completewe establish the Hahn-Banach theorem about the existence of continuous linearforms. This is then applied to obtain the first properties of the duality mapsinto the various forms of the linear bidual. In this section we encounter for thefirst time the phenomenon in nonarchimedean functional analysis that crucial

aspects of the theory depend on special properties of the nonarchimedean fieldK. The ultimate reason for this difficulty is that K need not to be locallycompact. A satisfactory substitute for compact subsets in locally convex K-vector spaces only exists if the field K is spherically complete. This will bediscussed systematically in 12 of the third chapter.

1 Nonarchimedean fields

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Let K be a field. A nonarchimedean absolute value on K is a function | | :K IR such that, for any a, b K we have

(i) |a| 0,

(ii) |a| = 0 if and only if a = 0,

(iii) |ab| = |a| |b|,

(iv) |a + b| max(|a|, |b|).

The condition (iv) is called the strict triangle inequality. Because of (iii) themap | | : K IR+ is a homomorphism of groups. In particular we have|1| = | 1| = 1. We always will assume in addition that | | is non-trivial, i.e.,that

(v) there is an a0 K such that |a0| = 0, 1.

It follows immediately that |n 1| 1 for any n ZZ. Moreover, if |a| = |b| forsome a, b K then the strict triangle inequality actually can be sharpened into

the equality|a + b| = max(|a|, |b|) .

To see this we may assume that |a| < |b|. Then |a| < |b| = |b + a a| max(|b+a|, |a|), hence |a| < |a+b| and therefore |b| |a+b| max(|a|, |b|) = |b|.

Via the distance function d(a, b) := |b a| the set K is a metric and hencetopological space. The subsets

B(a) := {b K : |b a| }

for any a K and any real number > 0 are called closed balls or simply ballsin K. They form a fundamental system of neighbourhoods of a in the metricspace K. Likewise the open balls

B (a) := {b K : |b a| < }

form a fundamental system of neighbourhoods of a in K. As we will see belowB(a) and B

(a) are both open and closed subsets of K. Talking about open

and closed balls therefore does not refer to a topological distinction but only tothe nature of the inequality sign in the definition.

We point out the following two simple facts.

1) If | | denotes the usual archimedean absolute value on IR then, for any b B (a), we have ||b| |a|| = ||(b a) + a| |a|| |max(|b a|, |a|) |a|| < .This means that the absolute value | | : K IR is a continuous function.

2) For b0 B (a0) and b1 B (a1) we have b0 + b1 B

(a0 + a1) and

b0b1 Bmax(,|a0|,|a1|)

(a0a1). The latter follows from b0b1 a0a1 = (b0

a0)(b1a1) +(b0 a0)a1+a0(b1a1). This says that addition + : KK Kand multiplication : K K K are continuous maps.

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Lemma 1.1:

i. B(a) is open and closed in K;

ii. if B(a) B(a) = then B(a) = B(a);

iii. If B and B are any two balls in K with B B = then either B B orB B;

iv. K is totally disconnected.

Proof: The assertions i. and ii. are immediate consequences of the strict triangleinequality. The assertion iii. follows from ii. To see iv. let M K be anonempty connected subset. Pick a point a M. By i. the intersection M B(a) is open and closed in M. It follows that M is contained in any ball arounda and therefore must be equal to {a}.

Clearly the assertions i.-iii. hold similarly for open balls. The assertion ii. says

that any point of a (open) ball can serve as its midpoint. On the other hand thereal number is not uniquely determined by the set B(a) and therefore cannotbe considered as the radius of this ball.

Another consequence of the strict triangle inequality is the fact that a sequence(an)nIN in K is a Cauchy sequence if and only if the consecutive distances|an+1 an| converge to zero if n goes to infinity.

Definition:

The field K is called nonarchimedean if it is equipped with a nonarchimedean

absolute value such that the corresponding metric space K is complete (i.e., everyCauchy sequence in K converges).

From now on throughout the book K always denotes a nonarchimedean fieldwith absolute value | |.

Lemma 1.2:

i. o := {a K : |a| 1} is an integral domain with quotient field K;

ii. m := {a K : |a| < 1} is the unique maximal ideal of o;

iii. o = o\m;

iv. every finitely generated ideal in o is principal.

Proof: The assertions i.-iii. again are simple consequences of the strict triangleinequality. For iv. consider an ideal a o generated by the finitely many ele-ments a1, . . . , am. Among the generators choose one, say a, of maximal absolutevalue. Then a = oa.

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The ring o is a valuation ring and is called the ring of integers of K. The fieldo/m is called the residue class field of K.

The reader should convince himself that, for any a o and any 1, the ballB(a) is an additive coset a + b for an appropriate ideal b o.

Examples:

1) The completion Qp ofQ with respect to the p-adic absolute value |a|p := pr

if a = pr mn such that m and n are coprime to the prime number p. The field Qpis locally compact.

2) The p-adic absolute value | |p extends uniquely to any given finite field ex-tension K ofQp. (Remember that there are plenty of such extensions since the

algebraic closure Qp is not finite over Q.) Any such K again is locally compact.

3) The completion Cp ofQp. This field is not locally compact since the set of

absolute values |Cp| is dense in IR+ (though countable).

4) The field of formal Laurent series C{{T}} in one variable over C with theabsolute value |

nZZ anT

n| := emin{n:an=0}. The ring of integers of this fieldis the ring of formal power series C[[T]] over C. Since C[[T]] is the infinite disjointunion of the open subsets a +TC[[T]] with a running over the complex numbersthe field C{{T}} is not locally compact.

The above examples show that the topological properties of the field K canbe quite different. It therefore may come as no surprise that there is in facta stronger notion of completeness. To explain this we consider any decreasingsequence B1 B2 . . . of balls in K. If K happens to be locally compactthen the intersection

nIN Bn, of course, is nonempty. For a general field K

there is the following additional condition which ensures the same. For anynonempty subset A K call d(A) := sup{|a b| : a, b A} the diameter of A.If we require in addition that the diameters d(Bn) converge to zero if n goes toinfinity then choosing points an Bn we obtain a Cauchy sequence (an)n whichhas to converge and whose limit has to lie in the intersection

nIN Bn. But in

general, without any further condition, this intersectionnIN Bn indeed can be

empty as the following construction shows.

Let the field be K = Cp. Fix any sequence (an)n in Cp which as a subset

is dense in Cp (e.g., one can take the algebraic closure Q of Q in Cp writtenas a sequence). In addition fix a sequence (n)n of real numbers such that1 > 1 > 2 > . . . >

12 . Consider now the equivalence relation on Cp defined by

a 1

b if |b a| 1. The equivalence classes clearly are balls. Since the value

group |Cp | = pQ is dense in IR+ their diameter is 1. Moreover there certainly is

more than one equivalence class. In particular, we may fix an equivalence classB1 such that a1 B1. Repeating this procedure with the equivalence relation

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on B1 defined by a 2

b if |b a| 2 we find a ball B2 B1 of diameter 2

such that a2 B2. Continuing with this construction we inductively obtain adecreasing sequence of balls B1 B2 . . . in Cp such that

d(Bn) = n and an Bn

for every n IN. We claim that the intersectionnIN Bn is empty. Otherwise

let b nIN Bn. We then have Bn = Bn(b) for any n IN and hence B1/2(b)

nIN Bn. As a consequence none of the an can be contained in the nonemptyopen subset B1/2(b). This contradicts the density of the sequence (an)n.

Definition:

The field K is called spherically complete if for any decreasing sequence of ballsB1 B2 . . . in K the intersection

nIN Bn is nonempty.

Any finite extension K ofQp is locally compact and hence spherically complete.On the other hand the field Cp, by the above discussion, is not sphericallycomplete.

Lemma 1.3:

Let K be spherically complete and let (Bi)iI be any family of balls in K suchthat Bi Bj = for any two i, j I; then

iIBi = .

Proof: Choose a sequence (in)nIN of indices in I such that d(Bi1) d(Bi2) . . .

and such that for every index i I there is a natural number n such that d(Bi) d(Bin). It follows from Lemma 1.1.iii that then Bi1 Bi2 . . . and that forany i I there is an n IN such that Bi Bin . Hence

iIBi =

nIN Bin is

nonempty.

Another important class of nonarchimedean fields is formed by those for whichthe value group |K| is a discrete subset ofIR+. These fields are called discretelyvalued. Examples of discretely valued fields K are finite extensions ofQp and thefield of Laurent series C{{T}}. The field Cp on the other hand is not discretelyvalued.

Lemma 1.4:

The subgroup |K| IR+ either is dense or is discrete; in the latter case thereis a real number 0 < r < 1 such that |K| = rZZ.

Proof: Let us assume that |K| is not dense in IR+. Then log |K| is not

dense in IR. Set := sup (log |K| (, 0)). We claim that actually is themaximum of this set. Otherwise there is a sequence 1 < 2 < . . . in log |K|

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which converges to . But then i i+1 is a sequence in log |K| (, 0)

converging to zero which implies that = 0. In this case we find for any > 0a log |K| such that < < 0. Consider now an arbitrary IRand choose an integer m ZZ such that m < (m 1). It follows that0 m < < and hence that log |K| is dense in IR which is a

contradiction. This establishes the existence of this maximum and consequentlyalso the existence of r := max(|K| (0, 1)). Given any s |K| there is anm ZZ such that rm+1 < s rm. We then have r < s/rm 1 which, by themaximality of r, implies that s = rm. This shows that |K| = rZZ.

Lemma 1.5:

The ring of integers o of a discretely valued field K is a principal ideal domain.

Proof: Let a o be an ideal. By the discreteness we find an a a such that|a| = max{|b| : b a}. Then a = ao.

Lemma 1.6:

Any discretely valued field K is spherically complete.

Proof: Let B1 B2 . . . be any decreasing sequence of balls in K. Then d(Bn)is a decreasing sequence of numbers in |K| which, by the discreteness, eitherbecomes constant (so that the intersection

nIN Bn even contains a ball) or con-

verges to zero (so that we know from our initial discussion that the intersectionnIN Bn is nonempty).

2 Seminorms

Let V be a K-vector space throughout this section. A (nonarchimedean) semi-norm q on V is a function q : V IR such that

(i) q(av) = |a| q(v) for any a K and v V,

(ii) q(v + w) max(q(v), q(w)) for any v, w V.

Since in the following exclusively nonarchimedean seminorms will appear wesimply speak of seminorms. Note that as an immediate consequence of (i) and(ii) one has:

- q(0) = |0| q(0) = 0,

- q(v) = max(q(v), q(v)) q(v v) = q(0) = 0 for any v V,

- |q(v) q(w)| q(v w) for any v, w V.

Moreover, with the same proof as before, one has

- q(v + w) = max(q(v), q(w)) for any v, w V such that q(v) = q(w).

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The vector space V in particular is an o-module so that we can speak abouto-submodules of V.

Definition:

A subsetA V is calledconvex if eitherA is empty or is of the formA = v+A0for some vector v V and some o-submodule A0 V.

Note that in the above definition the submodule A0 is uniquely determined bythe convex subset A. The following properties are immediately clear:

- If the convex subset A contains the zero vector then it is an o-submodule;

- if A is convex then v + A and b A are convex for any v V and any b K;

- if A and B are convex then so, too, is A + B = {v + w : v A, w B};

- the image as well as the preimage under a K-linear map of a convex subsetagain is convex.

Lemma 2.1:

Let (Ai)iI be a family of convex subsets in V; we then have:

i. the intersectioniIAi is convex;

ii. if for any two i, j I there is a third k I such that Ai Aj Ak then theunion

iIAi is convex.

Proof: i. We only need to consider the case where the intersection is nonempty.

Fix a vector v iIAi. Then Ai = v + Bi, for any i I, with some o-submodule Bi V. We therefore see that

iIAi = v +

iIBi is convex.

ii. Similarly we may assume that there is a vector v iIAi. Put J :=

{i I : v Ai}. By the assumption we are making in the assertion we haveiIAi =

iJAi. For i J we may write Ai = v +Bi with some o-submodule

Bi V. It follows thatiIAi = v +

iJBi. But again as a consequence of

our assumptioniJBi is, indeed, an o-submodule.

Definition:

A lattice L in V is an o-submodule which satisfies the condition that for anyvector v V there is a nonzero scalar a K such that av L.

In fact, for a lattice L V the natural map

Ko

L= V

a v av

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is a bijection. The surjectivity holds by definition. For the injectivity (whichholds for any o-submodule L V) consider any linear equation

ni=1 aivi = 0

with the vectors vi lying in L. Choose a K and bi o such that ai = abifor any 1 i n. In K o L we then obtain

i ai vi =

i a biv =

a (

ibiv) = a 0 = 0.

On the other hand a lattice in our sense does not need to be free as an o-module. The preimage of a lattice under a K-linear map again is a lattice. Asthe following argument shows the intersection L L of two lattices L, L Vagain is a lattice. Let v V and a, a K such that av L and av L.If a o then a1 o and hence v = (a1)av a1L L. We therefore mayassume that a, a o. Then aav L L.

For any lattice L V we define its gauge pL by

pL : V IRv inf

vaL|a| .

We claim that pL is a seminorm on V. First of all, for any b K and anyv V, we compute

pL(bv) = infbvaL

|a| = inf vb1aL

|a| = infvaL

|ba| = |b| infvaL

|a| = |b| pL(v) .

Secondly, the inequality pL(v + w) max(pL(v), pL(w)) is an immediate con-sequence of the following observation: For a, b K such that |b| |a| we haveaL + bL = aL.

On the other hand for any given seminorm q on V we define the o-submodules

L(q) := {v V : q(v) 1} and L(q) := {v V : q(v) < 1} .

We claim that L(q) L(q) are lattices in V. But, since we assumed theabsolute value | | to be non-trivial, we find an a K such that |an| convergesto zero if n IN goes to infinity. This means that for any given vector v V wefind an n IN such that q(anv) = |an| q(v) < 1.

Lemma 2.2:

i. For any lattice L V we have L(pL) L L(pL);

ii. for any seminormq on V we have co pL(q) q pL(q) where co := sup|b|


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This shows that pL(q) q. Moreover, if |b| < 1 then |b| infq(v)|a|

|a| < pL(q)(v).

It follows that there must be an a K such that q(v) |a| and |ba| < pL(q)(v).The latter inequality means that v baL(q) and hence that |b| |a| < q(v). Weobtain copL(q)(v) co|a| q(v).

3 Normed vector spaces

In this section we study a particular class of seminorms on a K-vector space V.

Definition:

A seminorm q on V is called a norm if

(iii) q(v) = 0 implies that v = 0.

Moreover, a K-vector space equipped with a norm is called a normed K-vectorspace.

It is the usual convention to denote norms by (and not by q). A normedvector space (V, ) will always be considered as a metric space with respect tothe metric d(v, w) := vw. It is therefore in particular a Hausdorff topologicalspace. Extending the language of the first section we introduce the closed balls(or simply balls)

B(v) := {w V : w v }

and the open balls

B (v) := {w V : w v < }

for any v V and any > 0. For a fixed v and varying each of them forma fundamental system of open neighbourhoods of v in V. It is immediatelyclear that B(0) and B (0) are lattices in V and that B(v) = v + B(0) andB (v) = v + B

(0) are convex subsets. As in the first section one shows:

1) Addition and scalar multiplication in V as well as the norm on V are con-tinuous maps. (The continuity of the former two maps is usually expressed bysaying that a normed vector space is a topological vector space.)

2) (Open) balls are open and closed subsets.

3) If the intersection of two balls B(v) and B(w) is nonempty then B(v) =B(w); a corresponding statement holds for open balls.

4) IfB and B are two (open) balls with nonempty intersection then B B orB B.

Definition:

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Proof: Let us first assume that the second assertion holds true. Consider anarbitrary sequence (vn)nIN in V converging to some vector v V. Then thesequence vn v converges to the zero vector and hence the norms vn vconverge to zero. It follows from our assumption that the norms f(vn) f(v)converge to zero as well. This implies that the sequence f(vn) converges to f(v)

and shows that f is continuous.

We now assume vice versa that f is continuous. There is then an > 0 such thatf1(B1(0)) B(0). Since the absolute value | | is non-trivial we may assume to be of the form = |a| for some a K. This means that f(v) 1 providedv |a|. Let now v be an arbitrary nonzero vector in V and choose an integerm ZZ such that |a|m+2 < v |a|m+1. We compute

f(v) = |a|m f(amv) |a|m < |a|2 v .

Corollary 3.2:

L(V, W) is a normed K-vector space with respect to the norm

f := sup{f(v)

v: v V\{0}} = sup{

f(v)

v: v V such that 0 < v 1}.

The above norm on L(V, W) is called the operator norm. We warn the readerthat since the set of values V may be different from the set of absolute values|K| we in general have

f(v) = sup{f(v) : v V such that v = 1}.

Proposition 3.3:

If W is a Banach space so, too, is L(V, W).

Proof: Let (fn)nIN be any Cauchy sequence in L(V, W). Then, in particular,fn is a Cauchy sequence in IR so that the limit lim

nfn exists. Moreover,

because of

fn+1(v) fn(v) = (fn+1 fn)(v) fn+1 fn v

fn(v), for any v V, is a Cauchy sequence in W. By assumption the limitf(v) := lim

nfn(v) exists in W. It is obvious that

f(av) = af(v) for any a K .

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For v, v V we compute

f(v) + f(v) = limn

fn(v) + limn

fn(v) = limn

(fn(v) + fn(v))

= limn

fn(v + v)

= f(v + v) .

This means that v f(v) is a K-linear map which we denote by f. Since

f(v) = limn

fn(v) ( limn

fn) v

it follows from Prop. 3.1 that f is continuous, i.e., that f L(V, W). Finallythe inequality

f fn = sup{

(ffn)(v)

v } = sup{

limm

fm(v)fn(v)

v }

supmn

fm+1 fm

shows that f indeed is the limit of the sequence (fn)n in L(V, W).

Corollary 3.4:

V := L(V, K) is a Banach space.

Definition:

V is called the dual Banach space to V.

We list two further simple properties:

1) The linear mapL(V, W) L(W, V)

f f() := f

is continuous satisfying f f (observe that f f).

2) The linear mapV V

v v() := (v)

is continuous satisfying v v.

It unfortunately turns out that in the nonarchimedean world there are nonzeronormed vector spaces whose dual Banach space is zero, i.e., which do not possess

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series =n=1 ((n)) 1(n) in co(X). Applying the continuous linear form

gives () =n=1 ((n)) (1(n)) and hence

|()|

supxX |(x) (1x)|

supxX

|(1x)| = supxX

|(x)| = .

Since was arbitrary it follows that . This shows that our map isisometric and a fortiori injective. For the surjectivity take any (X). Wedefine a linear form co(X) through

() :=xX

(x)(x) .

Its continuity is a consequence of the inequality |xX (x)(x)|

. We obviously have = .

4 Locally convex vector spaces

Let (Lj)jJ be a nonempty family of lattices in the K-vector space V such thatwe have

(lc1) for any j J and any a K there exists a k J such that Lk aLj,and

(lc2) for any two i, j J there exists a k J such that Lk Li Lj .

The second condition implies that the intersection of two convex subsets v + Li

and v+ Lj either is empty or contains a convex subset of the form w + Lk. Thismeans that the convex subsets v + Lj for v V and j J form the basis of atopology on V which will be called the locally convex topology on V defined bythe family(Lj). For any vector v V the convex subsets v + Lj , for j J, forma fundamental system of open and closed neighbourhoods of v in this topology.

Definition:

A locally convexK-vector space is aK-vector space equipped with a locally convextopology.

Lemma 4.1:

If V is locally convex then addition V V+

V and scalar multiplication KV

V are continuous maps.

Proof: For the continuity of the addition we only need to observe that (v +Lj)+(w + Lj) (v + w) + Lj . For the continuity of the scalar multiplication considerarbitrary elements a K, v V, and j J. Since Lj is a lattice we find a

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scalar b K such that bv Lj , and by (lc1) and (lc2) we find a k J suchthat aLk + bLk Lj. We then have (a + bo) (v + Lk) av + Lj .

Since on a nonzero K-vector space the scalar multiplication cannot be continuousfor the discrete topology we see that the discrete topology is not locally convex.In the following we want to discuss an alternative way to describe locally convextopologies with the help of seminorms.

Let (qi)iI be a family of seminorms on the K-vector space V. The topologyon V defined by this family (qi)iI, by definition, is the coarsest topology on Vsuch that

- all qi : V IR, for i I, are continuous, and

- all translation maps v + . : V V, for v V, are continuous.

For any finitely many norms qi1 , . . . , qir in the given family and any real number

> 0 we set

V(qi1 , . . . , qir ; ) := {v V : qi1 , . . . , qir(v) } .

Lemma 4.2:

V(qi1 , . . . , qir ; ) is a lattice in V.

Proof: Since V(qi1 , . . . , qir ; ) = V(qi1 ; ) . . . V(qir ; ) and since the intersec-tion of two lattices again is a lattice it suffices to consider a single V(qi; ). It isobviously an o-submodule. Choose an a K such that |a| . Then V(q

i; )

contains the lattice aL(qi) and therefore must also be a lattice.

Clearly the family of lattices V(qi1 , . . . , qir ; ) in V has the properties (lc1) and(lc2) and hence defines a locally convex topology on V.

Proposition 4.3:

The topology on V defined by the family of seminorms (qi)iI coincides withthe locally convex topology defined by the family of lattices {V(qi1 , . . . , qir ; ) :i1, . . . , ir I, > 0}.

Proof: Let T, resp. T, denote the topology defined by the seminorms, resp. bythe lattices. By the defining properties for T all the convex sets v + V(qi1 , . . . ,qir ; ) are open for T. This means that T is finer than T

. To obtain theequality of the two topologies it remains to show that T satisfies the two definingproperties for T. The translation maps are continuous in T by Lemma 4.1. Tocheck the continuity of the seminorm qi in T let (, ) IR be an open intervaland v0 q

1i (, ) be a vector. Ifqi(v0) > 0 we choose a 0 < < qi(v0). Because

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of qi(v0 + v) = qi(v0) for v V(qi; ) we then have v0 + V(qi; ) q1i (, ).

If, on the other hand, qi(v0) = 0 then we choose a 0 < < and obtain < 0 qi(v0 + v) qi(v) < for v V(qi; ) which again means thatv0 + V(qi; ) q

1i (, ).

We see in particular that the normed vector spaces of the previous section arelocally convex. The above result has the following converse.

Proposition 4.4:

A locally convex topology on V defined by the family of lattices (Lj)jJ can alsobe defined by the family of gauges (pLj )jJ.

Proof: Let T, resp. T, denote the topology defined by the lattices Lj, resp. bythe seminorms pj := pLj . Given an > 0 we fix an a K

such that |a| .It follows from Lemma 2.2.i that aLj V(pj ; ). Using the condition (lc2) we

deduce that V(pj ; ) is open for T. This implies, by Prop. 4.3, that T T.For the converse we fix a b K such that 0 < |b| < 1. Again from Lemma 2.2.iwe obtain that V(pj ; |b|) Lj which means that Lj is open for T and hencethat T T.

These two results together show that the concept of a locally convex topologyis the same as the concept of a topology defined by a family of seminorms. Wefinish this discussion with several useful observations belonging to this context.For the rest of this section we let V be a locally convex K-vector space.

Lemma 4.5:

Let L be a lattice in V and q be a seminorm on V; we then have:

i. The seminormq is continuous if and only if the lattice L(q), or equivalentlythe lattice L(q), is open in V;

ii. the lattice L is open in V if and only if its gauge pL is continuous.

Proof: i. Being the preimage under q of an open subset in IR0 the lattice L(q)

is open ifq is continuous. Furthermore, L(q) being a union of additive translatesof L(q) is open as soon as L(q) is open. Assuming finally that L(q) is openlet (, ) IR be an open interval and v0 q

1(, ) be a vector. At the endof the proof of Prop. 4.3 we have seen that there is then an a K such thatq1(, ) contains the open neighbourhood v0 + aL(q) of v0. This means thatq is continuous.

ii. As we have just seen ifpL is continuous then L(pL) is open. But L(pL) Lby Lemma 2.2.i so that L is open as well. If on the other hand L is open then,again by Lemma 2.2.i, the lattice L(pL) also is open. By assertion i., thisamounts to the continuity of pL.

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Lemma 4.6:

Assume that the topology of V is defined by the family of lattices (Lj)jJ, resp.by the family of seminorms (qi)iI; then the closure of {0} in V is the o-module

jJLj =

iIq

1i (0); in particular, the following assertions are equivalent:

i. V is Hausdorff;ii. for any nonzero vector v V there is a j J such that v Lj;

iii. for any nonzero vector v V there is an i I such that qi(v) = 0.

Proof: The lattices Lj are open and therefore closed. Hence we have {0} j Lj. If the vector v is not in {0} we find a lattice Lk such that 0 v + Lk and

consequently v Lk. This gives the equality {0} =j Lj . On the other hand,

by Prop. 4.3 the lattices V(qi1 , . . . , qir ; ) form a fundamental system of neigh-bourhoods of the zero vector. It follows that

i q1i (0) =

i, V(qi1 , . . . , qir ; ) =

j Lj. For the equivalence of the assertions i.-iii. one only has to observe in ad-dition that as a consequence of the translation invariance of any locally convextopology (Lemma 4.1) V is Hausdorff if and only if any nonzero vector can beseparated from the zero vector.

Remark 4.7:

Assume that the topology of V is defined by the family of seminorms (qi)iI.For any finite subset F I we may form the continuous seminorm qF :=maxiF

qi. The family (qF)F is a defining family for the topology on V which has

the additional property that the convex subsets v + V(qF; ) form a basis of the

topology.

Next we want to investigate the topological properties of convex subsets in V.

Lemma 4.8:

Let A V be a convex subset; we then have:

i. The closure A of A is convex;

ii. if A is not open then its interior is empty;

iii. if A is open then it is also closed;iv. if A is an open neighbourhood of the zero vector then A is a lattice.

Proof: We may assume that A is nonempty. By a translation we are furthermorereduced to the case that A is an o-submodule. As a consequence of Lemma 4.1the closure A also is an o-submodule and hence is convex. If A is open then, bythe definition of locally convex topologies, it must contain an open lattice andtherefore is a lattice as well; being the complement of a union of additive cosets

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ofA it is closed. Finally, ifv is a vector in the interior ofA then A must containv + L for some open lattice L V; being an o-module it then also contains Land therefore has to be open.

The convex hull of a subset S V is defined to be

Co(S) :=

{S A V : A is convex} .

By Lemma 2.1.i this is the smallest convex subset of V which contains S. Be-cause of Lemma 4.8.i we have

Co(S) = Co(S) .

Lemma 4.9:

For any subset S V we have:

i. If S is open then its convex hull Co(S) is open;

ii. Co(S) =

{S A V : A is convex and closed}.

Proof: i. IfS is empty then Co(S) is empty. Otherwise we may assume, bytranslation, that S contains the zero vector so that Co(S) is an o-submodule.Since S and hence Co(S) then contain an open lattice Co(S) is open. ii. It

follows from Lemma 4.8.i that Co(S) is convex and closed.

In a metric space and hence in a normed vector space it is clear what is meantby a bounded subset. It is of utmost importance that this concept can also beintroduced in locally convex vector spaces.

Definition:

A subset B V is called bounded if for any open lattice L V there is ana K such that B aL.

It is almost immediate that any finite set is bounded, and that any finite unionof bounded subsets is bounded. We leave it to the reader to check the following:Assume that the topology on V is defined by the family of seminorms (qi)iI.Then a subset B V is bounded if and only if sup

vBqi(v) < for any i I.

Lemma 4.10:

Let B V be a bounded subset; then the closure of the o-submodule of V gen-erated by B and a fortiori the convex hull Co(B) are bounded.

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required property. Step 3: It remains to show that, given an arbitrary norm p

on Kn, the identity map (Kn, p)id

(Kn, ) is continuous. According to Prop.3.1 we have to find a c > 0 such that

v c p(v) for any v Kn .

This will be achieved by induction with respect to n. The case n = 1 is obviouswith c := p(1). Applying the induction hypothesis to the restriction p|Kn1

we obtain a constant c1 > 0 such that v c1 p(v) for any v V :=Ke1 . . . Ken1 K

n. With (V, ) also (V, p) is complete. It follows thatV is closed in (Kn, p) which implies that

1 c2 := p(en)/ infvV

p(en v) < .

We set

c := max(c1c2, c2/p(en)) > 0 .

Let now w Kn be any vector and write w = v + ben with v V and b K.Since c > c1 it suffices to consider those w for which b = 0. In this case wecompute

p(w) = |b| p(b1v + en) |b| p(en) c12 = p(ben) c

12

and hence

p(v) = p(w ben) max(p(w), p(ben)) c2 p(w) .

Finally

w = max(v, |b|) max(c1, p(en)1) max(c11 v, |b|p(en))

cc12 max(p(v), p(ben)) c p(w) .

5 Constructions and examples

In this section we will discuss various general ways to construct locally convexvector spaces. Some of them will be illustrated by concrete examples.

A. Subspaces

Let V be a locally convex vector space and let U V be a vector subspace.Then the subspace topology of U induced by V is locally convex defined by alllattices LU where L runs over a defining family of lattices in V, or equivalentlyby all restrictions q|U where q runs over a defining family of seminorms on V.

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B. Quotient spaces

Let V be a locally convex vector space and let U V be a vector subspace.The quotient topology on V /U is locally convex defined by all lattices L + Uwhere L runs over a defining family of lattices in V. If one wants to describe

the quotient topology in terms of seminorms then one has to be a little careful.We first recall that for any seminorm q on V one has the quotient seminorm

q(v + U) := infuU

q(v + u) ;

it satisfies

L(q) + U = L(q) .

Let now (qi)iI be a defining family of seminorms for the topology of V. Usingthe above identity together with Remark 4.7 it follows that the quotient topology

on V /U is defined by the family of quotient seminorms (qF)F where F runs overthe finite subsets of I.

C. The finest locally convex topology

IfV is any K-vector space then the family of all lattices in V, or equivalently thefamily of all seminorms on V, defines a locally convex topology which obviouslyis the finest such topology on V. If V is equipped with the finest locally convextopology then any linear map from V into any other locally convex K-vectorspace is continuous. Moreover, any vector subspace U V is closed; in partic-

ular, V is Hausdorff. To see this choose vectors (vj)jJ such that (vj + U)j isa basis of V /U; then U is the intersection of the lattices Ln :=

j bnovj + U

where b K is a fixed scalar such that 0 < |b| < 1.

In particular, the uniquely determined locally convex and Hausdorff topologyon a finite dimensional vector space (Prop. 4.13) has to be the finest locallyconvex one.

D. Initial topologies

Let V be a K-vector space. Assume we are given a family (Vh)hH of locallyconvex K-vector spaces together with linear maps fh : V Vh. The coarsesttopology on V for which all the maps fh are continuous is called the initialtopology on V with respect to the family (fh)h. It is locally convex defined byall the lattices which are finite intersections of lattices in the family (f1h (Lhj))h,jwhere (Lhj)j is a defining family of lattices for the topology on Vh. Equivalentlyit is defined by the seminorms (qhi fh)h,i where (qhi)i is a defining family ofseminorms for Vh.

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A special case of this construction is the following. Let V =hHVh be the

direct product and let fh : V Vh be the projection maps. The correspondinginitial topology on V is called the direct product topology. We recall that in thissituation V is Hausdorff if and only if all the Vh are Hausdorff.

Example 1:

Let KIN :=nIN K be the countable direct product of one dimensional K-

vector spaces. This is an example of a locally convex and Hausdorff vectorspace whose topology cannot be defined by a single norm. Otherwise, by Cor.4.12, there should be a bounded open lattice L KIN. By the way the directproduct topology is constructed we may assume that L is of the form L =nF o

nIN\FK for some finite subset F IN. But the absolute value

on K viewed as a continuous seminorm on KIN via the projection to a factorcorresponding to some n F is not bounded on L.

Example 2:

Put X := Cp \Qp and let Oalg(X) denote the Qp-vector space of all Qp-rational

functions in one variable all of whose poles lie in Qp. We will construct acountable family of norms 1/n, for n IN, on O

alg(X) in the following way.For any n IN define

X(1/n) := {x X : |x| n and |x a| 1/n for any a Qp} .

We leave it to the reader to check that these sets X(1/n) are infinite.

Claim: R1/n

:= supxX(1/n)

|R(x)| < for any R Oalg(X).

Proof: Write

R =a0 + a1T + . . . + adT

dej=1(T bj)

with ai, bj Qp .

We then have numerator(R)1/n nd maxi(|ai|) and denominator(R)1/n

(1/n)e and hence R1/n nd+e maxi(|ai|).

This means that 1/n is a norm on Oalg(X). We define the Qp-Banach space

O1/n(X) to be the completion of the normed vector space (Oalg(X), 1/n).

Because of the inclusions X(1/n) X(1/(n + 1)) the identity maps (Oalg(X), 1/(n+1)) (O

alg(X), 1/n) are continuous and induce therefore continuouslinear maps

O1/(n+1)(X) O1/n(X) .

We define

O(X) := limn

O1/n(X)

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to be the corresponding projective limit equipped with the initial topology withrespect to the projection maps. The space O(X) can be viewed as a space ofcertain Cp-valued functions on X as follows.

Claim: X =

nIN X(1/n) .

Proof: Let x X be point and choose n0 IN such that |x| n0. We considerthe continuous function

Qp IR>0a |x a| .

We have |x a| = |a| > |x| if |x| < |a|. The subset {a Qp : |a| |x|} onthe other hand is compact so that its image in IR>0 under the above function isbounded. This function therefore is bounded below by 1/n1 for some n1 IN.It follows that x X(1/n) for n := max(n0, n1).

This result in particular means that, if F(Y, Cp) denotes the vector space of allCp-valued functions on a set Y, then we have F(X,Cp) = lim

n

F(X(1/n),Cp).

The inclusion

(Oalg(X), 1/n) BC(X(1/n),Cp)

is by definition an isometry. Since the space of bounded continuous Cp-valuedfunctions on the right hand side is a Banach space it extends to an isometry

O1/n(X) BC(X(1/n),Cp)

which in the limit gives rise to a Qp-linear embedding

O(X) limn

BC(X(1/n),Cp) limn

F(X(1/n),Cp) = F(X,Cp)

of O(X) into the space of all functions on X.

The space X = Cp \ Qp is called the p-adic upper half plane and the functionsin O(X) are called the rigid analytic or holomorphic functions on X.

E. Locally convex final topologies

Again let V be a K-vector space and (Vh)hH be a family of locally convex K-vector spaces. But this time we assume given linear maps fh : Vh V. Thenthere is a unique finest locally convex topology on V for which all the maps fhare continuous. It is called the locally convex final topology on V with respectto the family (fh)h, and it is defined by the family of all lattices L V suchthat f1h (L) is open in Vh for every h H. In general this topology is strictlycoarser than the finest topology on V making all the fh continuous.

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Lemma 5.1:

Assume that V carries the locally convex final topology with respect to a familyof linear maps fh : Vh V; we then have:

i. a K-linear map f : V W into some other locally convex K-vector space

W is continuous if and only if all the maps f fh : Vh W, for h H, arecontinuous;

ii. a seminorm q on V is continuous if and only if the seminorm q fh on Vhis continuous for any h H;

iii. assume that the topology onVh is defined by the family of lattices (Lhj)jJ(h)and that, in addition, we have V =

hHfh(Vh); then the topology on V is

defined by the family of lattices {hHfh(Lhj(h)) : j(h) J(h)}.

Proof: i. The other implication being trivial we assume that the maps f fh arecontinuous. To see that f is continuous it suffices to show that f1(M) is open

in V for any open lattice M W. By assumption (f fh)1(M) is open in Vh.Because of the obvious identity (f fh)

1(M) = f1h (f1(M)) this implies that

f1(M) is open.

ii. Again one implication being trivial we assume that the q fh are continuous.Using Lemma 4.5.i we see that on the one hand each L(q fh) is open in Vh andthat on the other hand it suffices to show that L(q) is open in V. But this isimmediate from the identity f1h (L(q)) = L(q fh).

iii. It is clear that the family of lattices in the assertion satisfies the conditions(lc1) and (lc2) and therefore defines a locally convex topology T on V. Becauseof f1h

0

(hHfh(Lhj(h))) Lh0j(h0) this topology T is coarser than the locallyconvex final topology. On the other hand, whenever L V is an open lattice wethen find, for any h H, a j(h) J(h) such that f1h (L) Lhj(h). It followsthat

hHfh(Lhj(h)) L. This shows that the two topologies in fact are equal.

In the following we want to look more closely at two special cases of this con-struction.

E1. The locally convex direct sum

Let (Vh)hH be a fixed family of locally convex K-vector spaces. We form thedirect sum V := hHVh and equip it with the locally convex final topologywith respect to the inclusion maps Vh0

hHVh. This locally convex vector

space V is called the locally convex direct sum of the Vh.

Lemma 5.2:

i. The inclusion maphHVh

hHVh is continuous;

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ii. if the set H is finite then the identity maphHVh

=hHVh is a topo-

logical isomorphism.

Proof: i. By the definition of the direct product topology the inclusions Vh0

hHVh are continuous. The assertion therefore follows from Lemma 5.1.i.

ii. This follows from the definition of the direct product topology and Lemma5.1.iii.

Lemma 5.3:

Let Uh Vh, for any h H, be a vector subspace with the subspace topology;we then have:

i. The locally convex direct sum topology onhHUh is the subspace topology

with respect to the inclusionhHUh

hHVh;

ii. the quotient vector space (hHVh)/(

hHUh) is the locally convex direct

sum of the quotients Vh/Uh;

iii. if Uh is closed in Vh for any h H thenhHUh is closed inhHVh.Proof: i. By Lemma 5.1.i the inclusion

h Uh

h Vh is continuous. On the

other hand, by Lemma 5.1.iii the locally convex direct sum topology onh Uh

is defined by the lattices of the formhMh where Mh is an open lattice in

Uh. Choose, for any h H, an open lattice Lh Vh such that Uh Lh Mh.

Putting Lh := Lh + Mh we have

hMh = (

h Uh) (

h Lh) which shows

that the left hand side is open in the subspace topology.

ii. According to the universal property of the quotient topology and Lemma5.1.i the bijection

(hHVh)/(hHUh) hHVh/Uh(h vh) + (

h Uh)

h(vh + Uh)

is continuous. Let, on the other hand, L be an open lattice in the left hand sideand denote its preimage in

h Vh, resp. its image in the right hand side, by L,

resp. by M. Then Lh := L Vh is an open lattice in Vh and Lh := (Lh+ Uh)/Uhis an open lattice in Vh/Uh. We therefore see that M contains the open latticeh Lh and hence is open.

iii. By Lemma 5.1.i all the projection maps prh0 :hVh Vh0 are continuous.

The assertion therefore is a consequence of the identityhH

pr1h (Uh) =hH

Uh .

In the proof of the first assertion of the above lemma we have used a simpleargument which will be used over and over again and which we therefore want

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to point out explicitly: Let W be a locally convex vector space and let U Wbe a subspace; for any open lattice M U one can find an open lattice L Vsuch that M = U L.

Corollary 5.4:

If Vh is Hausdorff for any h H thenhHVh is Hausdorff.

Proof: Apply Lemma 5.3.iii to the subspaces Uh := {0} and use Lemma 4.6.

E2. The strict inductive limit

Let V be a K-vector space and let

V1 V2 . . . V

be an increasing sequence of vector subspaces such that V = nIN Vn. Wemoreover assume that each Vn is equipped with a locally convex topology Tn insuch a way that

Tn+1|Vn = Tn for any n IN .

We equip V with the locally convex final topology T with respect to the in-clusions Vn V. In this situation V is called the strict inductive limit of theVn.

Proposition 5.5:

i. T |Vn = Tn for any n IN;

ii. if Vn is Hausdorff for any n IN then V is Hausdorff;

iii. if Vn is closed in Vn+1 for any n IN then Vn is closed in V for any n IN.

Proof: i. Fix an n IN and let Ln Vn be an open lattice. Because of theassumption that Tm+1|Vm = Tm we inductively find open lattices Ln+m Vn+msuch that Ln+m = Vn+m Ln+m+1 for any m 0. Then L :=

m0 Ln+m

clearly is a lattice in V. Moreover, L is open in V since L Vn+m = Ln+m forany m 0. In particular, Ln = Vn L which proves that Ln is open in thesubspace topology on Vn induced by V.

ii. Let v V be any nonzero vector. We have v Vm for some m IN. Since Vmis assumed to be Hausdorff we find an open lattice Lm Vm such that v Lm.The same inductive construction as under i. produces an open lattice L Vsuch that Lm = Vm L and hence v L. Using Lemma 4.6 we conclude thatV is Hausdorff.

iii. Fix an n IN and consider any vector v V \ Vn. We have v Vm forsome m > n. Since Vn is closed in Vm by assumption we find an open lattice

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Lm Vm such that (v + Lm) Vn = . Applying our inductive construction athird time there is an open lattice L V such that Lm = Vm L. It followsthat (v + L) Vn = ((v + L) Vm) Vn = (v + Lm) Vn = .

Proposition 5.6:

Assume thatVn is closed inVn+1 for anyn IN; then a subsetB V is boundedin V if and only if B Vm for some m IN and B is bounded in Vm.

Proof: We first assume that B is bounded in Vm. If L V is any open latticethen Vm L is an open lattice in Vm and we find an a K such that B a(Vm L) aL.

Now let B V be any bounded subset. Fix once and for all a scalar b K suchthat 0 < |b| < 1. Arguing by contradiction we assume that B is not containedin any Vn. We then find a sequence of natural numbers n1 < n2 < . . . and asequence (vk)kIN in B such that

() vk Vnk+1 \ Vnk for any k IN .

Note that the sequence (bkvk)k in V also satisfies the property (). We derive acontradiction in two steps.

Step 1: First we construct an open lattice L V which does not contain anyof the vectors bkvk. We start by fixing an open lattice L1 Vn1 and choosingan open lattice L2 Vn2 such that L1 = Vn1 L

2. With Vn1 also bv1 + Vn1 is

closed in Vn2 . Since bv1 + Vn1 does not contain the zero vector we find an openlattice L2 Vn2 such that L

2 L

2 and (bv1 + Vn1) L

2 = . The open lattice

L2 := L1 + L2 in Vn2 then satisfies

Vn1 L2 = L1 and bv1 L2 .

Repeating this construction we inductively obtain, for any k IN, an open latticeLk in Vnk satisfying

Vnk Lk+1 = Lk and bkvk Lk+1 .

Then L :=

kIN Lk is an open lattice in V not containing any b

kvk.

Step 2: We now show that given any open lattice L V we have bkvk L forany sufficiently big k IN. By Lemma 4.5.ii the gauge pL is continuous. SinceB is bounded there must be a sufficiently big c IN such that pL(vk) < |b|c

for any k IN. It follows, using Lemma 2.2.i, that bkvk L(pL) L for anyk c.

Since these two steps are in contradiction to each other there must be an m INsuch that B Vm. Let finally Lm Vm be an open lattice. Because of

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Prop. 5.5.i we find an open lattice L V such that Vm L = Lm. Thesubset B being bounded in V there is an a K such that B aL. We obtainB Vm aL = a(Vm L) = aLm. This proves that B is bounded in Vm.

Example 3:

Let X be a locally compact topological space. In section 3 we had discussed asan example the normed K-vector space (Cc(X), ). In the following we willconstruct another natural locally convex topology on Cc(X) which is finer thanthe norm topology.

Consider, for any compact subset A X, the vector subspace

CA(X) := { Cc(X) : |(X\ A) = 0} ;

equipped with the norm this is a Banach space. It is clear that Cc(X) is

the union of all this subspaces CA(X). We consider on Cc(X) the locally convexfinal topology with respect to the inclusions CA(X) Cc(X). By Lemma5.1.i the evaluation linear forms x() := (x) on Cc(X) are continuous. TheCA(X) =

xA ker(x) as subspaces ofCc(X) therefore are closed. In particular,

because of {0} = C(X), the locally convex vector space Cc(X) is Hausdorff.

Let us assume in addition that X is -compact, i.e., that X has a countablecovering by compact subsets. We then find an increasing sequence of compactsubsets A1 A2 . . . X such that X is covered by the interiors of the An.In particular, any compact subset of X is already contained in some Am. Sincethe inclusions CAn(X) CAn+1(X) are isometries it follows that Cc(X) is the

strict inductive limit of the increasing sequence of Banach spaces CAn(X). Wewill see later that therefore this new locally convex topology on Cc(X) is (incontrast to the norm topology) complete.

The space Cc(X) with this locally convex final topology is the starting pointof measure theory. The continuous linear forms on this space are called the(K-valued) Radon measures on X.

6 Spaces of continuous linear maps

In this section let V and W denote two locally convex K-vector spaces. Sincethe addition and the scalar multiplication in W are continuous, by Lemma 4.1,the continuous linear maps (or operators) from V into W form a vector subspace

L(V, W) := {f HomK(V, W) : f is continuous}

of the K-vector space HomK(V, W) of all linear maps. We will discuss in thissection a general technique to equip the vector space L(V, W) with a locally

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convex topology. As it turns out there are various ways to do this but whichall follow the same pattern. The following notion will play a crucial role in thisdiscussion.

Definition:

A subset H HomK(V, W) is called equicontinuous if for any open latticeM W there is an open lattice L V such that f(L) M for every f H.

It is obvious that any equicontinuous subset already is contained in L(V, W).

Proposition 6.1:

Suppose that the topology on V is defined by the family of seminorms (qi)iI; for a subsetH HomK(V, W) the following assertions are equivalent:

i. H is equicontinuous;

ii. for any continuous seminorm p on W there is a continuous seminorm q onV such that

p(f(v)) q(v) for any v V and f H ;

iii. for any continuous seminorm p on W there is a constant c > 0 and finitelymany i1, . . . , ir I such that

p(f(v)) c max(qi1

(v), . . . , qir(v)) for any v V and f H .

Proof: Let us first assume that H is equicontinuous. By definition we then havean open lattice L V such that f(L) L(p) for any f H. According toProps 4.3 and 4.4 we find an > 0 and finitely many i1, . . . , ir I such thatV(qi1 , . . . , qir ; ) L. We certainly may assume that 0 < = |b| < 1 for someb K. Hence

p(f(v)) 1 for any f H, provided max(qi1(v), . . . , qir(v)) |b| .

If max(qi1(v), . . . , qir(v)) = 0 then max(qi1(av), . . . , qir(av)) = 0 for any a K and therefore |a| p(f(v)) = p(f(av)) 1 for any a K. This implies

p(f(v)) = 0. If, on the other hand, max(qi1(v), . . . , qir(v)) > 0 then we maychoose an integer m such that |b|m+2 < max(qi1(v), . . . , qir(v)) |b|

m+1. Weobtain p(f(v)) = |b|m p(f(bmv)) |b|m < |b|2 max(qi1(v), . . . , qir(v)). Thismeans that the assertion iii. holds true with the constant c := |b|2. Theimplication from iii. to ii. is trivial by putting q := c max(qi1 , . . . , qir) .

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We finally assume that the assertion ii. holds true. Let M W be an openlattice. According to the Remark 4.7 we find an > 0 and a continuous semi-norm p on W such that V(p; ) M. By assumption we have a correspondingcontinuous seminorm q on V such that p(f(v)) q(v) for any f H. Thenf(V(q; )) V(p; ) M for any f H.

Corollary 6.2:

Suppose that the topology on V is defined by the family of seminorms (qi)iI;for an arbitrary seminorm q on V the following assertions are equivalent:

i. q is continuous;

ii. there is a constant c > 0 and finitely many i1, . . . , ir I such that

q(v) c max(qi1(v), . . . , qir(v)) for any v V .

Proof: Let W := V but equipped with the topology defined by the seminorm q.

The continuity ofq then amounts to the continuity of the identity map Vid

W.By Prop. 6.1 for H := {id} the latter is equivalent to the assertion ii.

If we use this corollary for the space W then it follows that in Prop. 6.1.iii thecontinuous seminorm p on W only needs to run over a defining family for thetopology.

Corollary 6.3:

Let H L(V, W) be an equicontinuous subset and let p be a continuous semi-norm on W; then q(v) := supfHp(f(v)) is a continuous seminorm on V.

The approach to define on L(V, W) a certain family of locally convex topologiesis based on the following two parallel observations. Fix a bounded subset B V.

1. For any open lattice M W the subset

L(B, M) := {f L(V, W) : f(B) M}

is a lattice in L(V, W). It is clear that L(B, M) is an o-submodule. If f L(V, W) is any continuous linear map then, by the boundedness ofB, there hasto be an a K such that B af1(M). This means that f(B) aM orequivalently that a1f L(B, M).

2. For any continuous seminorm p on W the formula

pB(f) := supvB

p(f(v))

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defines a seminorm on L(V, W). The only point to observe is that p f is acontinuous seminorm on V so that p(f(B)) is a bounded subset in IR0. Wehave the obvious identity

L(pB) = L(B, L(p)) .

Let now B be a fixed family of bounded subsets ofV. The locally convex topologyon L(V, W) defined by the family of seminorms {pB : B B, p a continuousseminorm on W} is called the B-topology. We write

LB(V, W) := L(V, W) equipped with the B-topology.

Lemma 6.4:

Assume that the family B is closed under finite unions; then the topology onLB(V, W) can be defined by the family of lattices {L(B, M) : B B, M W an open lattice}.

Proof: The family of lattices in question is nonempty and satisfies (lc1) and(lc2). We have L((pM)B) L(B, L(pM)) L(B, M) which shows that anylattice L(B, M) is open in LB(V, W). Let on the other hand p1,B1 , . . . , pr,Brbe finitely many of the defining seminorms for the B-topology and let a K.Setting B := B1 . . . Br and M := a(L(p1) . . . L(pr)) we have L(B, M) V(p1,B1 , . . . , pr,Br ; |a|).

Starting from the family B we may, using Lemma 4.10, define the usually muchlarger family B of all those bounded subsets B V for which there is an a Kand finitely many B1, . . . , Bm B such that aB is contained in the closure ofthe o-submodule generated by B1 . . . Bm. In particular, the family B alwaysis closed under finite unions.

Lemma 6.5:

The B- and B-topologies on L(V, W) coincide.Proof: For trivial reasons the

B-topology is finer than the B-topology. On the

other hand let M W be an open lattice and B B. Choose a K andB1, . . . , Bm B such that aB is contained in the closure of the o-submodulegenerated by B1 . . . Bm. Then a[L(B1, M) . . . L(Bm, M)] L(B, M).

Lemma 6.6:

If W is Hausdorff and ifBB B generates a dense vector subspace in V then

LB(V, W) is Hausdorff.

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Proof: We check the condition ii. in Lemma 4.6. Let 0 = f L(V, W). Byassumption there is a B B and a vector v B such that f(v) = 0. Since,moreover, W is assumed to be Hausdorff there is an open lattice M W suchthat f(v) M. It follows that f L(B, M).

Examples:

1) Let B be the family of all one point subsets of V. The corresponding B-topology is called the weak topology or the topology of pointwise convergence.We write Ls(V, W) := LB(V, W). The weak topology in fact is the initial topol-ogy with respect to the evaluation maps

L(V, W) Wf f(v)

for v V.

2) Let B be the family of all bounded subsets in V. The corresponding B-topology is called the strong topology or the topology of bounded convergence.We write Lb(V, W) := LB(V, W).

If W is Hausdorff both locally convex vector spaces Ls(V, W) and Lb(V, W) areHausdorff.

Remark 6.7:

If V and W are normed vector spaces then the topology on Lb(V, W) is definedby the operator norm .

Proof: Let B denote the family of all bounded subsets in V and let Bo have

the unit ball B1(0) in V as its single member. Since B = Bo it follows fromLemma 6.5 that the topology on Lb(V, W) is defined by the norm f :=supv1f(v). We trivially have f

f. Fix a b K such that 0 1. Since anLn L(q) there is a vectorvn Ln such that q(vn) > |a|

n. Hence the seminorm q is not bounded on thesequence of vectors (vn)n. But by construction this sequence converges to thezero vector and therefore is bounded. This is a contradiction.

We now come to the second class of locally convex vector spaces. Here thestarting point is the fact that any open lattice in V also is closed.

Definition:

A locally convex vector space V is called barrelled if every closed lattice in V isopen.

Proposition 6.15: (Banach-Steinhaus)

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If V is barrelled then any bounded subset H Ls(V, W) is equicontinuous.

Proof: Let M W be an open lattice and consider the o-submodule L :=fHf

1(M) of V. We have to show that L is open. Since L obviously isclosed it suffices to check that L is a lattice. Let v V be any vector. By theboundedness of H in Ls(V, W) there is an a K such that H aL({v}, M).This implies that a1v L.

As with the above proposition we sometimes follow the convention to namevery basic results the same way their counterparts over the real or complexfield are named. This does not mean, e.g., that Prop. 6.15 was proved byBanach-Steinhaus. But it helps a lot to remember the content of this result.

Later (in 12) we will see that over a spherically complete field the Banach-Steinhaus theorem actually characterizes barrelled vector spaces.

Corollary 6.16:

Suppose that V is barrelled; then in LB(V, W), for any B-topology which is finerthan the weak topology, the bounded subsets coincide with the equicontinuoussubsets.

Proof: This is Lemma 6.8 together with Prop. 6.15.

Examples:

1) If V has no countable covering V =

nIN An by closed subsets An with

empty interior then V is barrelled.Proof: Let L V be a closed lattice and fix an a K such that |a| > 1. ThenV =

nIN a

nL. By assumption there must therefore exist an n IN such thatanL and consequently L has a nonempty interior. This means we find a vectorv L and an open lattice L V such that v +L L. But then L also containsL and hence is open.

2) If V is metrizable and is complete with respect to a defining metric (e.g., Vis a Banach space) then V is barrelled.

Proof: By Baires theorem ([B-GT] Chap.IX 5.3 Thm.1) we are in the situationof 1).

3) Suppose that the topology on V is the locally convex final topology withrespect to a family of linear maps fh : Vh V; if all the Vh are barrelled thenso, too, is V.

Proof: This is obvious.

4) Quotient spaces, locally convex direct sums, and strict inductive limits ofbarrelled vector spaces are barrelled.

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Proof: These are special cases of 3).

7 Completeness

In this section we will discuss the concepts of completeness and completion fora general locally convex K-vector space V. We recall that a directed set (I, )is a set I together with a partial order which has the additional property thatfor any two elements i, j I there is a third element k I such that i k and

j k.

Definition:

a) A net (vi)iI in V is a family of vectors vi in V where the index set I isdirected;

b) a net (vi)iI is said to converge to the vector v V if for any open latticeL V there is an index i I such that vj v L for any j i; we also say inthis case that this net is convergent;

c) a net (vi)iI is called a Cauchy net if for any open lattice L V there is anindex i I such that vj vk L for any j, k i;

d) a subset A V is called complete if every Cauchy net in A converges to avector in A.

Remark 7.1:

i. Any (convergent, Cauchy) sequence is a (convergent, Cauchy) net;

ii. any convergent net is a Cauchy net;

iii. if V is Hausdorff then a net in V converges to at most one vector;

iv. any closed subset of a complete subset is also complete;

v. if V is Hausdorff then any complete subset of V is closed;

vi. suppose that the topology on V is defined by the family of seminorms (qj)jJ;then a net(vi)iI inV converges to the vectorv V if and only if(qj(viv))iIconverges to zero for any j J;

vii. letf : V W be a continuous linear map between locally convex K-vectorspaces; if(vi)iI converges to v V, resp. is a Cauchy net, then(f(vi))iI con-verges to f(v), resp. is a Cauchy net.

Remark 7.2:

If V is metrizable then we have:

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i. V is complete if and only if every Cauchy sequence in V is convergent;

ii. suppose that the topology on V can be defined by a translation invariant metricd (i.e., d(v + w, v + w) = d(v, v) for any v, v, w V); then V is complete ifand only if the metric space (V, d) is complete.

Proof: The second assertion is an immediate consequence of the first one and theidentity d(v, w) = d(v w, 0). For the first assertion only the reverse implicationhas to be considered. Let (vi)iI be a Cauchy net in V. Since V is metrizablethere is a decreasing sequence L1 L2 . . . of open lattices in V which forma fundamental system of neighbourhoods of the zero vector. For any n IN wechoose an index in I such that vjvk Ln for any j, k in. We certainly mayassume that i1 i2 . . . Write wn := vin ; then (wn)n is a Cauchy sequence. Byassumption it converges to a vector v V. We claim that the original Cauchynet (vi)i also converges to v. Let L V be any open lattice. It contains someLm, and we have vj v {vj wn}nm Lm L = L for j im.

In particular, any Banach space is complete.

Lemma 7.3:

Let Vo V be a dense vector subspace; any continuous (w.r.t. the subspacetopology) linear map fo : Vo W into a complete Hausdorff locally convexK-vector space W extends uniquely to a continuous linear map f : V W.

Proof: For the uniqueness part of the assertion we may assume that fo = 0. Thenf(V) = f(Vo) fo(Vo) = {0} = {0} which amounts to f = 0. For the existenceof f let denote the set of all open lattices in V; it is directed by the reverse ofthe inclusion relation. We define f(v) for a given vector v V as follows. SinceVo is dense in V there is, for any L , a vector vL Vo such that vLv L. Byconstruction the net (vL)L converges to v and therefore is a Cauchy net. Hence(fo(vL))L is a Cauchy net in W and converges, by assumption, to a vectorwhich we denote by f(v). We proceed in three steps. Step 1: The vector f(v)does not depend on the choice of the net (vL)L. Let (vL)L be another choice.The net (vL vL)L then converges to the zero vector in Vo. The continuityof fo implies that the net (fo(vL) fo(vL))L converges to the zero vector inW. This shows that v f(v) is a well defined map on V which extendsfo. Step 2: The map f is K-linear. Let (vL)L and (wL)L be nets as above

converging to the vectors v and w, respectively. Given any two scalars a, b Kthe net (fo(avL + bwL))L = (afo(vL) + bfo(wL))L converges to af(v) + bf(w)in W. On the other hand, the net (avL + bwL)L converges to av + bw. Thereindexed net (avcL + bwcL)L where c K such that |c1| = max(|a|, |b|, 1)has the property used above for the definition of f(av + bw). It follows thataf(v) + bf(w) = f(av + bw). Step 3: The map f is continuous. Let M Wbe an open lattice. By the continuity of fo there is an open lattice L

V suchthat L Vo = f1o (M). For any v L

we can choose a net (vL)L as above

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converging to v so that this net in fact lies in L Vo. Then (fo(vL))L is a netin M. Since M is closed it is complete. We obtain that f(v) M and hencethat f(L) M.

Remark 7.4:

Let Vo V be a dense vector subspace equipped with the subspace topology; wehave:

i. Any continuous seminorm qo on Vo extends uniquely to a continuous semi-norm q on V, and any continuous seminorm on V arises in this way;

ii. if Lo runs over all open lattices in Vo then the closure Lo in V runs over allopen lattices in V and satisfies Lo Vo = Lo.

Proof: i. Using the inequality |qo(v) qo(w)| qo(v w) for any v, w Vo

the existence and uniqueness proof for q is entirely analogous to the argumentfor Lemma 7.3. Moreover, any continuous seminorm on V, by uniqueness, isthe extension of its restriction to Vo. i i . If L is an open lattice in V then wehave, by the density of Vo, that L L Vo. But since L is closed we also haveL Vo L. It remains to remark that given Lo there is an open lattice L Vsuch that L Vo = Lo.

Proposition 7.5:

For any locally convex K-vector space V there exists an up to a unique topo-

logical isomorphism unique complete Hausdorff locally convex K-vector space Vtogether with a continuous K-linear map cV : V V such that the followinguniversal property holds true: For any continuous K-linear map f : V Winto a complete Hausdorff locally convex K-vector space W there is a unique

continuous K-linear map f : V W such that f = f cV . We moreoverhave:

i. The image im(cV) is dense in V;ii. the map cV induces a topological isomorphism between V /{0} with the quo-tient topology and im(cV) with the subspace topology.

Proof: The unicity statement immediately follows from the universal property.For the existence proof we may assume V to be Hausdorff. Let again denotethe set of all open lattices in V directed by the reverse of the inclusion relation.

As an o-module we define V as the projective limitV := lim

L

V /L

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of the o-module quotients V /L. But V in fact is a K-vector space since multi-plication by any a K induces an automorphism of . Since V is Hausdorffthe obvious K-linear map

cV : V

V

v (v + L)L

is injective. For any open lattice M V we define the o-submodule M of V byM := lim

L,LM

M/L .

An equivalent characterization is given by

M = {(vL + L)L

V : vM M} .

This construction has the following straightforward properties:

1) M is a lattice in V.2) We have c1V (

M) = M.3) The intersection of all M is equal to {0}.4) The family of all lattices M is nonempty and satisfies (lc1) and (lc2).The property 4) allows us to equip V with the locally convex topology definedby the family of all the lattices

M. Because of 3) this topology is Hausdorff,

and because of 2) the map cV induces a topological isomorphism between V and

im(cV). Next we check that im(cV) is dense in V; in fact, the obvious formulav cV(vL) L for anyv = (vL+ L)L V shows more generally that M is densein M.The existence and uniqueness of the map f is now an immediate consequenceof Lemma 7.3. It remains to check that V is complete. Let (vi)iI with vi =(vi,L+L)L be a Cauchy net inV. We then find, for any L , an index i(L) Isuch that vj vk L, i.e., vj,L vk,L L for any j, k i(L). It follows thatthe vector

v := (vi(L),L+ L)L V is well defined and satisfies

v

vi L for any

i i(L) and any L. This means, of course, that the Cauchy net (

vi)i converges

tov.In the above proof we have seen that the open lattice M is the closure in V ofthe open lattice M V. We therefore obtain from Remark 7.4.ii that the Mconstitute all the open lattices in V.Definition:

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V is Hausdorff by Cor. 5.4. It therefore suffices to show that the canonical map

cV : V V is surjective. For any family = (Lh)hH of open lattices Lh Vhwe have the open lattice L :=

hHLh in V. According to Lemma 5.1.iii the

completion is the projective limit

V = lim

V /L; the map cV corresponds to the

obvious map

V =hH

Vh =hH

( lim

LhVh

Vh/Lh) lim

=(Lh)

(hH

Vh/Lh) = V .Let now (

h vh + Lh) with v

h Vh be a vector in the right hand side. Each

set of indices H() := {h H : vh Lh} is finite. We note that the cosetvh + Lh only depends on Lh and not on the family to which Lh belongs. Thisvector therefore lies in the left hand side if we can show that the union

H()

still is finite. If this union would be infinite there would have to exist a sequence

of pairwise different indices (hn)nIN

and families n = (Ln,h)hH such thatvnhn Ln,hn for every n IN. Consider in this case the family = (Lh)h withLh = Ln,hn for h = hn and Lh = Vh for h = h1, h2, . . . By construction thecorresponding set of indices H() = {h1, h2, . . .} would be infinite which leadsto a contradiction.

Lemma 7.9:

Let V be the strict inductive limit of the vector subspaces V1 V2 . . .; if Vnis complete for every n IN then V is complete, too.

Proof: Let (vi)iI be a Cauchy net. In a first step we show that there is anm IN such that for any i I and any open lattice L V there is a j i suchthat vj Vm + L. Assume that, for any k IN, there is an open lattice Lk Vand an i(k) I such that

vj Vk + Lk for all j i(k) .

We certainly may assume that the L1 L2 . . . are decreasing. Consider theopen lattice

L := Co(nIN

Vn Ln)

in V. We claim that L Vk + Lk for any k IN. It suffices to show thatVn Ln Vk + Lk for any n and k. But if n k then Vn Vk, and if n kthen Ln Lk. It follows that Vk + L Vk + Lk for any k IN. Choose now anindex i I such that vi1 vi2 L for any i1, i2 i. Letting k IN be such thatvi Vk we arrive at the contradiction that vj Vk + L Vk + Lk for any j i.

Denoting by , as usual, the set of all open lattices in V we introduce the setI directed by the partial order (i, L) (j,M) if i j and M L. Fix a

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natural number m with the property which we have established above. For anypair (i, L) I we then have an index i(L) i and a vector v(i,L) Vm suchthat v(i,L) vi(L) L.

In the next step we show that (v(i,L))(i,L)I in fact is a Cauchy net in Vm. Anyopen lattice in Vm is of the form Vm L for some L . Fix an i I such thatvk vl L for any k, l i. Consider now any two pairs (k, M), (l, N) (i, L).We have v(k,M) vk(M) M and v(l,N) vl(N) N. Since k

(M) k i,l(N) l i, and M + N L we obtain v(k,M) v(l,N) = (v(k,M) vk(M)) +(vk(M) vl(N)) + (vl(N) v(l,N)) M + L + N L.

Since Vm by assumption is complete the Cauchy net (v(i,L))(i,L) converges tosome vector v Vm. We conclude the proof by showing that the original Cauchynet (vi)i in V converges to the same vector v. Let L V be any open lattice.We find a pair (k, M) I such that

(1) v(l,N) v L for any (l, N) (k, M), and

(2) vk1 vk2 L for any k1, k2 k.

As a special case of (1) we have v(k,ML) v L. Since, by construction,v(k,ML) vk(ML) M L it follows that vk(ML) v L. Using (2) wefinally obtain that vl v L for any l k.

There is the following important weakening of the concept of completeness.

Definition:

A locally convex vector space V is called quasi-complete if every bounded closedsubset of V is complete.

Obviously every complete V is quasi-complete.

Lemma 7.10:

Every Cauchy sequence in V is contained in a bounded and closed subset.

Proof: Let B = {vn : n IN} be a Cauchy sequence in V. Then (q(vn))n,for any continuous seminorm q on V, is a Cauchy sequence in IR and hence is

bounded. It follows that B is bounded and that its closure B is bounded andclosed.

Proposition 7.11:

If V is quasi-complete then we have:

i. Every Cauchy sequence in V is convergent;

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ii. if V is metrizable then it is complete.

Proposition 7.12:

i. A closed vector subspace U of a quasi-complete locally convex vector space V

is quasi-complete (in the subspace topology);ii. the direct product V =

hHVh of a family of quasi-complete locally convex

vector spaces Vh is quasi-complete;

iii. the locally convex direct sum V =hHVh of a family of quasi-complete

and Hausdorff locally convex vector spaces Vh is quasi-complete;

iv. the strict inductive limit V of an increasing sequence V1 V2 . . . of closedand quasi-complete vector subspaces Vn is quasi-complete.

Proof: The assertion i. is obvious. The assertion iv. is a consequence of Prop.5.6. For ii. and iii. let prh : V Vh denote the projection maps. If B is a

bounded subset of V then B is contained in h prh(B), resp. in h prh(B).In the situation of ii. this latter set obviously is complete as a direct productof complete subsets. The same holds true in the situation of iii. once we estab-lish the fact that prh(B) = 0 for all but finitely many h H. Reasoning bycontradiction let us assume that there is a sequence of pairwise different indices(hn)nIN in H and a sequence of vectors (vn)nIN in B such that prhn(vn) = 0for any n IN. Fix a b K such that |b| > 1. Since Vhn is Hausdorff we findan open lattice Lhn Vhn such that prhn(b

nvn) Lhn. Define Lh := Vh forany h = h1, h2, . . . Then L :=

h Lh, by Lemma 5.1.iii, is an open lattice in

V. Since B is bounded we have B bmL for some m IN. This leads to thecontradiction that prh

n

(vn) bmLhn bnLhn for all n m.

Proposition 7.13:

Suppose that W is Hausdorff and quasi-complete; if the B-topology is finer thanthe weak topology then any equicontinuous closed subset H LB(V, W) is com-plete.

Proof: In a first step we consider the case of the weak topology. By Lemma 6.8the subset H is bounded in Ls(V, W) and hence in Map(V, W), and by Lemma6.10 it is closed in Map(V, W). It therefore suffices to show that Map(V, W)is quasi-complete. Let (fi)iI be a Cauchy net in a bounded closed subset B

of Map(V, W). Since the evaluation maps are continuous the subsets B(v) :={f(v) : f B} of W, for any v V, are bounded and closed. By our assump-tions on W the Cauchy net (fi(v))i in B(v) converges to a uniquely determined

vector f(v) B(v) W. This defines a map f Map(V, W). By constructionthe Cauchy net (fi)i converges to f. Since B was closed we in fact have f B.

We now turn to the general case. According to Lemma 6.10 the closure H ofH in Ls(V, W) is equicontinuous. By the case treated above H therefore is a

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complete subset of Ls(V, W). Since H is closed in the B-topology it sufficesto show that H is complete in LB(V, W) as well. We will in fact establish thefollowing more general assertion:

Let B and B be two families of bounded subsets of V such that the B-topologyis finer than the B-topology and the latter is finer than the weak topology; anysubset A L(V, W) which is complete for the B-topology also is complete forthe B-topology.

Let (fi)iI be a Cauchy net in A with respect to the B- and hence also withrespect to the B-topology. In the B-topology it converges, by assumption, tosome f A. To check that f is the limit in the B-topology as well let B B and M W be an open lattice. There is an i I such that fj fk L(B, M), or equivalently fj fk + L(B, M), for any j, k i. But L(B, M) =vB L({v}, M) is closed in the weak and hence in the B-topology. It follows

that f fk + L(B, M), or equivalently fk f L(B, M), for any k i.

Corollary 7.14:

IfV is barrelled andW is Hausdorff and quasi-complete thenLB(V, W) is Haus-dorff and quasi-complete for anyB-topology which is finer than the weak topology.

Proof: Lemma 6.6, Cor. 6.16, and Prop. 7.13.

Corollary 7.15:

If V is bornological and W is Hausdorff and quasi-complete then Lb(V, W) isHausdorff and quasi-complete.

Proof: Lemma 6.6, Prop. 6.12, and Prop. 7.13.

Proposition 7.16:

IfV is bornological andW is Hausdorff and complete thenLb(V, W) is Hausdorffand complete.

Proof: By Lemma 6.6 the space Lb(V, W) is Hausdorff. Let (fi)iI be a Cauchynet in Lb(V, W) and therefore in Ls(V, W) Map(V, W). This latter vectorspace visibly is Hausdorff and complete. Our net consequently has a point-wise limit f Map(V, W). In the proof of Lemma 6.10 we have seen thatHomK(V, W) is closed in Map(V, W). Hence f is K-linear. In order to show

that f indeed is the limit in Lb(V, W) of our Cauchy net we need three interme-diate assertions.

Claim 1: Let B V be a bounded subset and M W be an open lattice; thereis an i I such that (fk f)(B) M for any k i.

We certainly find an i I such that fj fk L(B, M) for any j, k i. Since{g Map(V, W) : g(B) M} is closed our claim follows by the same argumentwhich we have used at the end of the proof of Prop. 7.13.

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Claim 2: The map f respects sequences converging to the zero vector.

Let (vn)nIN be such a sequence in V and let M W be an open lattice. ByLemma 7.10 the subset B := {vn : n IN} is bounded. Applying our firstclaim we find an i I such that fi(vn) f(vn) M for any n IN. Since fiis continuous the sequence (fi(vn))n converges to the zero vector in W. Thismeans that there exists an m IN such that fi(vn) M for any n m. Itfollows that f(vn) M for any n m.

Claim 3: The map respects bounded subsets.

Otherwise there exists a bounded subset B V, an open lattice M W,and a scalar a K with |a| > 1 such that f(B) anM for any n IN. Inother words, for any n IN there is a vector vn B such that anf(vn) M.Since B is bounded we find, given an open lattice L V, an m IN such thatB amL. Because of anvn = (a1)nm(amvn) L for any n m we seethat the sequence (anvn)n in V converges to the zero vector. Applying our

second claim we arrive at a contradiction.Returning to the main line of the proof we now apply the last claim and Prop.6.13 to the K-linear map f and obtain that f is continuous. The first claimthen shows that the Cauchy net (fi)i converges to f in Lb(V, W).

We finish this section by looking at the situation where W is arbitrary but Vis quasi-complete. But first we have to introduce the following construction ofgeneral importance. Let A be any o-submodule in the locally convex K-vectorspace V. We let VA denote the vector subspace of V generated by A. SinceA is a lattice in VA the gauge pA is defined as a seminorm on VA. We always

view VA as equipped with the locally convex topology defined by pA. Since Ais bounded in VA it is a necessary condition for the inclusion VA

V to be

continuous that A is bounded in V.

Lemma 7.17:

If B is a bounded o-submodule in V then we have:

i. The inclusion VB

V is continuous;

ii. if V is Hausdorff then (VB , pB) is a normed vector space;

iii. if V is Hausdorff and B is complete then (VB , pB) is a Banach space.

Proof: i. Let L V be an open lattice. Since B is bounded there is an a K

such that B aL and hence that a1B VB L. This shows that VB L isan open lattice in VB.

ii. Assume that v VB is a vector such that 0 = pB(v) = inf vaB

|a|. It follows

that Kv L(pB) B. By the boundedness of B this implies that Kv is

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contained in any open lattice in V. Since V is Hausdorff we therefore must havev = 0.

iii. Let (vn)nIN be a Cauchy sequence in VB . In particular, there is an m INsuch that vj vk B for any j, k m. Fix a 0 = a o such that avn Bfor any n m. Then (avn)n is a Cauchy sequence in B VB and hence, byi., in B V. By assumption this sequence (avn)n converges to some vectorv B V. Fix a scalar c K such that 0 < |c| < 1. Since (avn)n is a Cauchysequence in VB there is an increasing sequence n1 < n2 < . . . of natural numberssuch that

pB(avj avk) |c|i+1 for any j, k ni and any i IN .

In particular,

avn1 + cB avn2 + c2B . . . avni + c

iB . . .

With B all the convex subsets avni + ciB are closed in V. It follows that the

limit v in V of the sequence (avni)i lies in the intersectioni avni + c

iB. Weconsequently have pB(v avni) |c|

i for any i IN. This shows that thesequence (avni)i converges to v already in VB . Our original Cauchy sequence(vn)n therefore converges in VB to a

1v.

Proposition 7.18:

Suppose tha

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Schneider-Nonarchimedean Functional Analysis 2