ńņš. 47 |

Bishop called his version of radical analysis ā˜Constructive Analysisā™ and

the reader will ļ¬nd an excellent account of it in the ļ¬rst few chapters of [7].

The ļ¬rst serious attempt to found a radical analysis was due to Brouwer who

called it ā˜Intuitionismā™.

Appendix I

Miscellany

This appendix consists of short notes on various topics which I feel should

be mentioned but which did not ļ¬t well into the narrative structure of this

book.

Compactness When mathematicians generalised analysis from the study of

metric spaces to the study of more general ā˜topological spacesā™, they needed a

concept to replace the Bolzano-Weierstrass property discussed in this book.

After some experiment, they settled on a property called ā˜compactnessā™.

It can be shown that a metric space has the property of compactness

(more brieļ¬‚y, a metric space is compact) when considered as a topologi-

cal space if and only if it has the Bolzano-Weierstrass property. (See Ex-

ercises K.196 and K.197 if you would like to know more.) We showed in

Theorem 4.2.2 that a subset of Rn with the standard metric has the Bolzano-

Weierstrass property if and only if it is closed and bounded.

Thus, if you read of a ā˜compact subset E of Rn ā™, you may translate

this as ā˜a closed and bounded subset of Rn ā™ and, if you read of a ā˜com-

pact metric spaceā™, you may translate this as ā˜a metric space having the

Bolzano-Weierstrass propertyā™. However, you must remember that a closed

bounded metric space need not have the Bolzano-Weierstrass property (see

Exercise 11.2.4) and that, for topological spaces, the Bolzano-Weierstrass

property is not equivalent to compactness.

Abuse of language The language of mathematics is a product of history as

well as logic and sometimes the forces of history are stronger than those of

logic. Logically, we need to talk about the value sin x that the function sin

takes at the point x, but, traditionally, mathematicians have talked about the

function ā˜sin xā™. To avoid this problem mathematicians tend to use phrases

like ā˜the function f : R ā’ R given by f (x) = x2 ā™ or ā˜the map x ā’ x2 ā™. In

421

422 A COMPANION TO ANALYSIS

Section 13.3, when we wish to talk about the map x ā’ L(Ī», x) = t(x)ā’Ī»f (x)

we write ā˜L(Ī», ) = t ā’ Ī»f ā™. (Many mathematicians dislike leaving a blank

space in a formula and use a place holder ā˜Ā·ā™ instead. They write ā˜L(Ī», Ā·) =

t(Ā·) ā’ Ī»f (Ā·)ā™.)

To a 19th century mathematician and to most of my readers this may

appear unnecessary but the advantage of the extra care appears when we need

to talk about the map x ā’ 1. However, from time to time, mathematicians

revert to their traditional habits.

Bourbaki calls such reversions ā˜abuses of language without which any

mathematical text runs the risk of pedantry, not to say unreadability.ā™

Perhaps the most blatant abuse of language in this book concerns se-

quences. Just as f (x) is not a function f , but the value of f at x, so an is

not the sequence

a1 , a 2 , a 3 , a 4 . . . ,

but the nth term in such a sequence. Wherever I refer to ā˜the sequence

an ā™, I should have used some phrase like ā˜the sequence (an )ā ā™. Perhaps

n=1

future generations will write like this, but it seemed to me that the present

generation would simply ļ¬nd it distracting.

Non-uniform notation When Klein gave his lectures on Elementary Mathe-

matics from An Advanced Standpoint [28] he complained

There are a great many symbols used for each of the vector op-

erations and, so far, it has proved impossible to produce a gener-

ally accepted notation. A commission was set up for this purpose

at a scientiļ¬c meeting at Kassel (1903). However, its members

were not even able to come to a complete agreement among them-

selves. None the less, since their intentions were good, each mem-

ber was willing to meet the others part way and the result was to

bring three new notations into existence1 ! My experience in such

things inclines me to the belief that real agreement is only possi-

ble if there are powerful economic interests behind such a move.

. . . But there are no such interests involved in vector calculus and

so we must agree, for better or worse, to let every mathematician

cling to the notation which he ļ¬nds most convenient or ā“ if he is

dogmatically inclined ā“ the only correct one.

1

In later editions Klein recorded the failure of similar committees set up at the Rome

International Mathematical Congress (1908).

423

Please send corrections however trivial to twk@dpmms.cam.ac.uk

Most mathematics students believe that there should be a unique notation

for each mathematical concept. This is not entirely reasonable. Consider the

derivative of a function f : R ā’ R. If I am interested in the slope of the

dy

curve y = f (x) it is natural to use the suggestive Leibniz notation . If I

dx

am interested in the function which is the derivative of f it is is more natural

to consider f , and if I am interested in the operation of diļ¬erentiation rather

than the functions it operates on I may well write Df . Diļ¬erent branches of

mathematics may have genuine reasons for preferring diļ¬erent notations for

the same thing.

Even if she disagrees, the student must accept that diļ¬erent notations

exist and there is nothing that she can do about it. A quick visit to the

library reveals the following notations for the same thing (here f : R3 ā’ R

is a well behaved function) :-

ā‚3f

, fxxz , fxxz , Dxxz f, f,113 , f113 , f113 , D113 f, D(2,0,1) f, D(2,0,1) f

2 ā‚z

ā‚x

and several others.

Here are three obvious pieces of advice.

(1) If you are writing a piece of mathematics and several notations exist

you must make it clear to the reader which one you are using.

(2) If you are reading a piece of mathematics which uses an unfamiliar

notation try to go along with it rather than translating it back into your

favourite notation. The advantage of a familiar notation is that it carries you

along without too much thought, the advantage of an unfamiliar notation is

that it makes you think again ā” and, who knows, the new notation might

turn out to better than the old.

(3) Do not invent a new notation when a reasonably satisfactory one

already exists.

Left and right derivatives In many ways, the natural subsets of Rm to do

analysis on are the open sets U , since, given a function f : U ā’ Rn and a

point x ā U we can then examine the behaviour of f ā˜in all directions round

xā™, on a ball

B(x, ) = {y : y ā’ x < },

for some, suļ¬ciently small, > 0.

However, as we saw in Theorem 4.3.1 and Theorem 4.5.5, continuous

functions behave particularly well on closed bounded sets. This creates a

certain tension, as the next exercise illustrates.

424 A COMPANION TO ANALYSIS

Exercise I.1. (i) Show that the map Rm ā’ R given by x ā’ x is contin-

uous.

(ii) If E is a closed bounded set, show that there exists an e0 ā E such

that e0 ā„ x for all x ā E. Give an example to show that e0 need not be

unique.

(iii) Show that the only subset of Rm which is both open and closed and

bounded is the empty set.

This tension is often resolved by considering functions deļ¬ned on an open

set U , but working on closed bounded subsets of U . In the special case, when

we work in one dimension and deal with intervals, we can use a diļ¬erent trick.

Deļ¬nition I.2. Let b > a and consider f : [a, b] ā’ R. We say that f has

right derivative f (a+) at a if

f (a + h) ā’ f (a)

ā’ f (a+)

h

as h ā’ 0 through values h > 0.

Exercise I.3. Deļ¬ne the left derivative f (bā’), if it exists.

When mathematicians write that f is diļ¬erentiable on [a, b] with deriva-

tive f they are using the following convention.

Deļ¬nition I.4. If f : [a, b] ā’ R is diļ¬erentiable at each point of (a, b) and

has right derivative f (a+) at a and left derivative f (bā’) at b, we say that

f is diļ¬erentiable on [a, b] and write f (a) = f (a+), f (b) = f (bā’).

Observe that this is not radically diļ¬erent from our other suggested ap-

proach.

Lemma I.5. If f : [a, b] ā’ R is diļ¬erentiable on [a, b] we can ļ¬nd an every-

Ė Ė

where diļ¬erentiable function f : R ā’ R with f (t) = f (t) for t ā [a, b].

Proof. Set

Ė

f (t) = f (a) + f (a)(t ā’ a) for t < a,

Ė for a ā¤ t ā¤ b,

f (t) = f (t)

Ė

f (t) = f (b) + f (b)(t ā’ b) for b < t.

Ė

It is easy to check that f has the required properties.

Exercise I.6. If f : [a, b] ā’ R is diļ¬erentiable on [a, b] with continuous

Ė

derivative, show that we can ļ¬nd an everywhere diļ¬erentiable function f :

Ė

R ā’ R with continuous derivative such that f (t) = f (t) for t ā [a, b].

425

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If f : [a, b] ā’ R is diļ¬erentiable on [a, b] and f : [a, b] ā’ R is diļ¬eren-

tiable on [a, b], it is natural to say that f is twice diļ¬erentiable on [a, b] with

derivative f = (f ) , and so on.

Exercise I.7. If f : [a, b] ā’ R is twice diļ¬erentiable on [a, b] with continuous

second derivative, show that we can ļ¬nd an everywhere twice diļ¬erentiable

Ė Ė

function f : R ā’ R with continuous second derivative such that f (t) = f (t)

for t ā [a, b]. Generalise this result.

Piecewise deļ¬nitions Occasionally mathematicians deļ¬ne functions as piece-

wise continuous, piecewise continuously diļ¬erentiable and so on. The under-

lying idea is that the graph of the function is made up of a ļ¬nite number of

well behaved pieces.

Deļ¬nition I.8. A function f : [a, b] ā’ R is piecewise continuous if we can

ļ¬nd

a = x0 < x1 < Ā· Ā· Ā· < x n = b

and continuous functions gj : [xjā’1 , xj ] ā’ R such that f (x) = gj (x) for all

x ā (xjā’1 , xj ) and all 1 ā¤ j ā¤ n

Deļ¬nition I.9. A function f : [a, b] ā’ R is piecewise linear (respectively

diļ¬erentiable, continuously diļ¬erentiable, inļ¬nitely diļ¬erentiable etc.) if it

is continuous and we can ļ¬nd

a = x0 < x1 < Ā· Ā· Ā· < x n = b

such that f |[xjā’1 ,xj ] : [xjā’1 , xj ] ā’ R is linear (respectively diļ¬erentiable, con-

tinuously diļ¬erentiable, inļ¬nitely diļ¬erentiable etc.) for all 1 ā¤ j ā¤ n.

Notice that Deļ¬nition I.8 does not follow the pattern of Deļ¬nition I.9.

Exercise I.10. (i) Show by means of an example that a piecewise continuous

function need not be continuous.

(ii) Show that a piecewise continuous function is bounded.

Exercise I.11. (This is a commentary on Theorem 8.3.1.) (i) Show that

any piecewise continuous function f : [a, b] ā’ R is Riemann integrable.

(ii) Show that, if f : [a, b] ā’ R is continuous on (a, b) and bounded on

[a, b], then f is Riemann integrable.

The deļ¬nitions just considered are clearly rather ad hoc. Numerical ana-

lysts use a more subtle approach via the notion of a spline (see, for example

Chapters 18 and onwards in [42]).

Appendix J

Executive summary

The summary is mainly intended for experts but may be useful for revision.

It may also be more useful than the index if you want to track down a

particular idea. Material indicated ā™„ . . . ā™„ or ā™„ā™„ . . . ā™„ā™„ is not central

to the main argument. Material indicated [ . . . ] is in appendices or exercises;

this material is either not central to the main argument or is such that most

students will have met it in other courses.

Introduction to the real number system

Need for rigorous treatment of analysis (p. 1). Limits in R, subsequences,

sums and products (p. 3). Continuity of functions from R to R (p. 7). The

real numbers R form an ordered ļ¬eld obeying the fundamental axiom that

every increasing bounded sequence converges (p. 9). Axiom of Archimedes

(p. 10). [Decimal expansion (Exercise 1.5.12, p. 13).] The intermediate value

theorem, proof by lion hunting (p. 14). [Countability (Appendix B, p. 383).

The real numbers are uncountable (Exercise 1.6.7, p. 17). Cantorā™s proof of

the existence of transcendentals (Exercise B.7, p. 385). Explicit construction

of a transcendental number (Exercise K.12, p. 435).] Diļ¬erentiation and the

mean value inequality (one dimensional case) (p. 18). Intermediate value

theorem equivalent to fundamental axiom (p. 22). ā™„ā™„Further informal

discussion of the status of the fundamental axiom (p. 25).ā™„ā™„

Equivalents of the fundamental axiom

Supremum, existence for bounded non-empty sets equivalent to fundamental

axiom, use as proof technique (p. 31). Theorem of Bolzano-Weierstrass,

equivalent to fundamental axiom, use as proof technique (p. 37).

Higher dimensions

Rm as an inner product space, Cauchy-Schwarz and the Euclidean norm

427

428 A COMPANION TO ANALYSIS

(p. 43). Limits in Rm (p. 46). Theorem of Bolzano-Weierstrass in Rm (p. 47).

Open and closed sets (p. 48). Theorem of Bolzano-Weierstrass in the context

of closed bounded subsets of Rm (p. 49). Continuity for many dimensional

spaces (p. 53). The image of a continuous function on a closed bounded

subset of Rm is closed and bounded, a real-valued continuous function on a

closed bounded subset of Rm is bounded and attains its bounds (p. 57). [The

intersection of nested, non-empty, closed, bounded sets is non-empty (Exer-

cise 4.3.8, p. 59).] Rolleā™s theorem and the one dimensional mean value theo-

rem (p. 60). Uniform continuity, a continuous function on a closed bounded

subset of Rm is uniformly continuous (p. 64).

Sums

(This material is treated in Rm .) General principle of convergence (p. 66).

Absolute convergence implies convergence for sums (p. 69). Comparison test

(p. 70). Complex power series and the radius of convergence (p. 71). ā™„Ratio

test and Cauchyā™s condensation test (p. 70). Conditional convergence, al-

ternating series test, Abelā™s test, rearrangement of conditionally convergent

series (p. 78). Informal discussion of the problem of interchanging limits

(p. 81). Dominated convergence theorem for sums, rearrangement of abso-

lutely convergent series, Fubiniā™s theorem for sums (p. 84). The exponential

function mainly for R but with mention of C, multiplication of power series.

(p. 91). The trigonometric functions, notion of angle (p. 98). The logarithm

on (0, ā), problems in trying to deļ¬ne a complex logarithm (p. 102). Powers

(p. 109). Fundamental theorem of algebra (p. 113).ā™„

Diļ¬erentiation from Rn to Rm

Advantages of geometric approach, deļ¬nition of derivative as a linear map,

Jacobian matrix (p. 121). Operator norm, chain rule and other elementary

properties of the derivative (p. 127). Mean value inequality (p. 136). Simple

local and global Taylor theorems in one dimensions, Cauchyā™s example of

a function with no non-trivial Taylor expansion, Taylor theorems depend

on the fundamental axiom (p. 141). Continuous partial derivatives imply

diļ¬erentiability, symmetry of continuous second order derivatives, informal

treatment of higher order local Taylor theorems, informal treatment of higher

order derivatives as symmetric multilinear maps (p. 146). Discussion, partly

informal, of critical points, hill and dale theorem (p. 154).

Riemann integration

Need for precise deļ¬nition of integral and area, Vitaliā™s example (p. 169).

Deļ¬nition of the Riemann integral via upper and lower sums, elementary

properties, integrability of monotonic functions (p. 172). Integrability of

429

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continuous functions, fundamental theorem of the calculus, Taylorā™s theorem

with integral form of remainder (p. 182). ā™„ Diļ¬erentiation under the inte-

gral for ļ¬nite range, Euler-Lagrange equation in calculus of variations, use

and limitations, Weierstrass type example (p. 190).ā™„ Brief discussion of the

Riemann integral of Rm -valued functions, fā¤ f (p. 202).

ā™„ Class of Riemann integrable functions not closed under pointwise

convergence (p. 205). Informal discussion of improper Riemann integration

(p. 207). Informal and elementary discussion of multiple integrals (no change

of variable formula), Fubini for continuous functions on a rectangle (p. 212),

Riemann-Stieltjes integration (p. 217). Rectiļ¬able curves and line integrals,

Schwarzā™s example showing the problems that arise for surfaces (p. 224).ā™„

Metric spaces

ā™„ Usefulness of generalising notion of distance illustrated by Shannonā™s

theorem on the existence of good codes (p. 233).ā™„ Metric spaces, norms,

limits, continuity, open sets (p. 241). All norms on a ļ¬nite-dimensional

space are Lipschitz equivalent (p. 246). Continuity of functions between

normed spaces (p. 251). ā™„ Informal discussion of geodesics illustrated by

the PoincarĀ“ metric on the upper half plane (p. 254).ā™„

e

Complete metric spaces

Deļ¬nition of completeness, examples of complete and incomplete metric

spaces including among incomplete ones the L1 norm on C([a, b]) (p. 263).

Completeness and total boundedness, equivalence of the conjunction of these

properties with the Bolzano-Weierstrass property (p. 272).

Uniform metric is complete, restatement of result in classical terms (uni-

form limit of continuous functions is continuous, general principle of uni-

form convergence) (p. 275). Uniform convergence, integration and diļ¬erenti-

ation, restatement for inļ¬nite sums, diļ¬erentiation under an inļ¬nite integral

(p. 282). Local uniform convergence of power series, power series diļ¬eren-

tiable term by term, rigorous justiļ¬cation of power series solution of diļ¬eren-

tial equations (p. 288). ā™„ An absolutely convergent Fourier series converges

to the appropriate function (p. 298).ā™„

Contraction mapping theorem

Banachā™s contraction mapping theorem (p. 303). Existence of solutions of

diļ¬erential equations by Picardā™s method (p. 305). ā™„ Informal discussion

of existence and non-existence of global solutions of diļ¬erential equations

(p. 310). Greenā™s function solutions for second order linear diļ¬erential equa-

tions (p. 318).ā™„

The inverse function theorem (p. 329). ā™„ The implicit function theorem

430 A COMPANION TO ANALYSIS

(p. 339). Lagrange multipliers and Lagrangian necessary condition (p. 347).

Lagrangian suļ¬cient condition, problems in applying Lagrange multiplier

methods (p. 353).ā™„

Completion of metric spaces

Density, completion of metric space, inheritance of appropriate structures

such as inner product (p. 355). Proof of existence of completion (p. 362).

ā™„ Informal discussion of construction of Z from N, Q from Z, C from R

(p. 364). Construction of R from Q (p. 369).ā™„ ā™„ā™„ Rapid, optimistic and

informal discussion of foundational issues (p. 375).ā™„ā™„

Appendix K

Exercises

At an elementary level, textbooks consist of a little explanation (usually sup-

plemented by a teacher) and a large number of exercises of a routine nature.

At a higher level the number of of exercises decreases and the exercises be-

come harder and more variable in diļ¬culty. At the highest level there may be

no exercises at all. We may say that such books consist of a single exercise:-

read and understand the contents. Because I would be happy if students

treated my text in this manner I have chosen to put most of the exercises in

an appendix.

I suspect that readers will gain most by tackling those problems which

interest them. To help them make a choice, I have labeled them in the

following manner [2.1, P]. The number 2.1 tells you that Section 2.1 may

be relevant, and the letters have the meanings given below. Like many similar

labeling systems it is not entirely satisfactory.

ā‘ Follows on from the preceding question.

ā‘ā‘ Follows on from an earlier question.

S Rather shorter or easier than the general run of questions.

M Methods type question. Forget theoretical niceties and concentrate on

getting an answer.

M! Just try and get an answer by fair means or foul.

T This question leads you through a standard piece of theory.

T! This question leads you through a standard piece of theory but in a

non-standard way.

P Problem type question.

G Uses general background rather than material in this book.

H The result of this exercise is not standard.

H! The result of this exercise is highly non-standard. Only do this exercise

if you are really interested.

431

432 A COMPANION TO ANALYSIS

Figure K.1: Apostolā™s construction.

ā

Exercise K.1. (irrationality of 2.) [1.1, G, T] The reader presumably

knows the classic proof that the equation n2 = 2m2 has no non-zero integer

solutions (in other words, x2 = 2 has no solution in Q). Here are two others.

(i) Show that, if n2 = 2m2 , then

(2m ā’ n)2 = 2(n ā’ m)2 .

Deduce that, if n and m are strictly positive integers with n2 = 2m2 , we

can ļ¬nd strictly positive integers n and m with n 2 = 2m 2 and n < n.

Conclude that the equation n2 = 2m2 has no non-zero integer solutions.

(ii) Our second argument requires more thought but is also more powerful.

We use it to show that, if N is a positive integer which is not a perfect square,

then the equation x2 = N has no rational solution.

To this end we suppose that x is a positive rational with x2 = N . Explain

why we can ļ¬nd a least positive integer m such that mx is an integer and

why we can ļ¬nd an integer k with k + 1 > x > k. Set m = mx ā’ mk and

show that m is an integer, that m x is an integer and that m > m ā„ 1. The

required result follows by contradiction. (This argument and its extensions

are discussed in [3].)

(iii) Apostol gave the following beautiful geometric version of the argu-

ment of part (i) (see Figure K.1). It will appeal to all fans of Euclidean

geometry and can be ignored by everybody else.

ńņš. 47 |