G12MAN Mathematical Analysis - MathsNet: School of ... · Why do we do all these proofs? Rigorous...

G12MAN Mathematical Analysis

Based on notes by Dr J. Zacharias

Modified by Dr J. F. Feinstein

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About these notes

These notes include all of the definitions and theorems of

the module.

Most of them are examinable.

The notes omit a number of proofs and explanations of

examples given during the lectures.

There are gaps left where these should be filled in by you

during the lectures. Writing this material yourself is

an important part of studying this module.

Technology permitting, Dr Feinstein’s annotated slides

from the lectures will be made available from the module

web page.

The entire 2009-10 version of G12MAN (including

annotated slides and videos) can be downloaded from the

web. See the G12MAN Module Lecture Notes page

for details.

Downloading this material is not an adequate

substitute for attending classes.

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Examinable and non-examinable material

Certain material in this module will be clearly marked as

Not examinable as bookwork, or NEB for short.

This does not mean that this material is irrelevant to

exam questions.

Some sections of some of the exam questions are likely to

be ‘unseen’.

It is possible that some of the NEB material could be

relevant to some unseen portions of exam questions.

For more information on the style of exam questions, see

the G12MAN Module Information page and the

G12MAN Module Feedback page.

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Why do we do all these proofs?

Rigorous arguments and justification are essential

throughout mathematics, and not just in ‘Pure’

Mathematics.

Even in those modules where you do not justify all your

reasoning in full, you may well be told that certain facts

are true but that the proof is beyond the scope of that

module.

Here are some examples to illustrate the need for rigour,

and, in particular, some applications of Mathematical

Analysis.

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• Dr Matthews tells me that in G12MDE in traffic

flow modelling and fluid dynamics, you need to

‘differentiate under the integral sign’ in a setting

where it would require mathematical analysis in order

to justify this manipulation.

• This interchange between differentiation and

integration also turns up in Professor Wood’s

module G13INF Statistical Inference.

Professor Wood says

‘This interchange is used to prove important

results in statistics such as the Cramer–Rao

lower bound.

In the module I do not provide rigorous

justification of this interchange though I say a

bit about when it can break down.’

Gap to fill in

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Additionally:

• Dr Wilkinson tells me that advanced mathematical

analysis (measure theory) is needed

‘. . . in order to do real world things (such as date

the primate origins and the human-chimp split).’

• Dr Louko used several theorems from mathematical

analysis in his latest Mathematical Physics paper.

• Professor Fyodorov tells me that if it wasn’t for the

fact that we sometimes have

limX→0

limY→∞

f(X,Y ) 6= limY→∞

limX→0

f(X,Y )

then there would be no permanent magnets!

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Here is an interesting example to think about that

illustrates some of the dangers.

For which constants α ≥ 0 is it true that

limn→∞

∫ 1

0

nαxn(1− x)dx

exists, and what value (possibly depending on α) does

the limit take in these cases?

Gap to fill in

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1 Introduction to Rd

In this chapter we look at subsets of

Rd = {(x1, x2, . . . , xd) | x1, x2, . . . , xd ∈ R} and

establish some of the basic notation and terminology that

we will need in the rest of the module.

You have met R1 = R, R2, R3 and to some extent Rd

already, for instance when solving larger systems of linear

equations.

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1.1 Set notation

We use the standard set theoretical notations from

G11ACF. In particular we denote the set of

• positive integers by N = {1, 2, 3, 4, . . . };

• integers by Z = {. . . ,−2,−1, 0, 1, 2, . . . };

• rational numbers (fractions) by

Q = {ab| a ∈ Z, b ∈ N};

• real numbers by R;

• non-negative real numbers by

R+ = {x ∈ R | x ≥ 0} = [0,∞).

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Subsets are often specified by a certain condition shared

by all its elements.

For instance certain subsets of R can be written as

{x ∈ R | x2 < x}

or

{x ∈ R | x2 + 1 = 0}.

The former is another way to write

{x ∈ R | 0 < x < 1}.

The latter is another way to write the empty set ∅.

Many different looking conditions or properties define the

same set.

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A set A is a subset of a set B, written A ⊆ B, if each

element in A lies also in B, i.e. x ∈ A implies x ∈ B (we

can write: x ∈ A⇒ x ∈ B).

Thus A = B if and only if A ⊆ B and B ⊆ A.

For sets C and D their intersection is

C ∩D = D ∩ C = {x | x ∈ C and x ∈ D}

= {x | x ∈ D, x ∈ C}

and their union is

C ∪D = D ∪ C = {x | x ∈ D or x ∈ C}.

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Similarly if D1, D2, . . . , Dn are sets then

D1 ∩D2 ∩ · · · ∩Dn =n⋂i=1

Di

= {x | x ∈ D1, x ∈ D2, . . . , x ∈ Dn}

and

D1 ∪D2 ∪ · · · ∪Dn =n⋃i=1

Di

= {x | x ∈ D1 or x ∈ D2 or . . . or x ∈ Dn}

Intersections and unions can also be defined for infinitely

many Di’s, see later.

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The difference between C and D is

C\D = {x ∈ C | x /∈ D}.

where x /∈ D means that x is not an element of D.

Note that this is often different from

D\C = {x ∈ D | x /∈ C} .

If A is a subset of a larger set B then its complement (in

B) is Ac = B\A.

Usually it is clear from the context what B is.

Examples: Complement of a subset of R.

Gap to fill in

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1.2 Cartesian products

The Cartesian product of two sets A and B is the set

of all ordered pairs of elements in A and B, i.e.

A×B = {(a, b) | a ∈ A, b ∈ B}.

(a1, b1) = (a2, b2) if and only if a1 = a2 and b1 = b2.

Example 1.2.1 R× R is the set of ordered pairs of real

numbers.

Each (x, y) ∈ R× R can be visualised as a point in a

coordinate system.

Gap to fill in

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The Cartesian product of two subsets A,B ⊆ R can be

visualised as well, e.g. if A = [1, 2] and B = [2, 4], then

A×B is represented by

Gap to fill in

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We can produce a variety of subsets of R2 by taking

products of subsets of R, but there are many others such

as S = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ x} or

R = {(x, y) | x2 + y2 = 1} which can not be written as

Cartesian products.

(See question sheets for details.)

Gap to fill in

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R× R is also denoted by R2.

Similarly A×B × C is the set of ordered triples

A×B × C = {(a, b, c) | a ∈ A, b ∈ B, c ∈ C}

etc.

We denote R× R× R by R3 and more generally

Rd = R× R× · · · × R︸ ︷︷ ︸d

= {(x1, x2, . . . , xd) | x1, x2, . . . , xd real numbers} ,

the set of all d-tuples of real numbers.

Here d is a positive integer. R1 is identified with R.

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1.3 Revision of Rd

Elements of Rd may be regarded as points which are

positioned in a certain way or as vectors which we can

add or multiply.

The former is Rd as an analytic or geometrical object,

the latter is Rd as a vector space: an algebraic object.

We will not worry about the distinction in this module.

Thus elements in Rd are denoted by (for example)

x,y, z,p, q,

etc. where x = (x1, x2, . . . , xd), y = (y1, y2, . . . , yd) etc.

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Sometimes we will write them as d-dimensional column

vectors.

x =

x1

x2...

xd

, y =

y1

y2...

yd

, z =

z1

z2...

zd

etc.

Column vectors may be added

x+ y =

x1 + y1

x2 + y2...

xd + yd

Column vectors may be multiplied by scalars λ ∈ R

λx = λ

x1

x2...

xd

=

λx1

λx2...

λxd

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Since we identify (x1, x2, . . . , xd) and

x1...

xd

we may

well also write

(x1, x2, . . . , xd) + (y1, y2, . . . , yd)

= (x1 + y1, x2 + y2, . . . , xd + yd)

etc.

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Recall the standard inner product on Rd:

〈x,y〉 =

⟨x1...

xd

,y1...

yd

⟩

=

d∑i=1

xiyi.

The norm corresponding to it is the Euclidean norm

defined by

‖x‖ =√〈x,x〉 =

(d∑i=1

|xi|2) 1

2

, for x =

x1...

xd

and may be regarded as the length of a vector.

Recall the Cauchy–Schwarz inequality

|〈x,y〉| ≤ ‖x‖ ‖y‖

which holds for all x,y ∈ Rd.

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Proposition 1.3.1 The function || · || : Rd → R+ has the

following properties, for all x and y in Rd and all λ ∈ R:

(i) ‖x+ y‖ ≤ ‖x‖+ ‖y‖ (triangle inequality);

(ii) ‖λx‖ = |λ|‖x‖ (homogeneity);

(iii) ‖x‖ = 0⇐⇒ x = 0.

The statement and applications of this result are

examinable as bookwork.

The proof of this result is NEB (not examinable as

bookwork), but is available on request.

Gap to fill in

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The three properties above are all you usually need to

know about the Euclidean norm ‖x‖ in this module.

Using our Euclidean norm ‖ · ‖, we define the Euclidean

distance between points x and y by

d(x,y) = ‖x− y‖ = ‖y − x‖ = d(y,x).

If z is another point in Rd we have

d(x,y) ≤ d(x, z) + d(z,y)

(triangle inequality for the Euclidean distance), since

Gap to fill in

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Throughout this module, the Euclidean distance

d(x,y) = ‖x− y‖

=√

(x1 − y1)2 + · · ·+ (xd − yd)2

will play an important role.

In this module, unless otherwise specified, when we

refer to distance in Rd we will always mean the

Euclidean distance.

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