ńņš. 19 |

ļ£¬B ļ£· ļ£¬ļ£·

ļ£¬ ļ£· = M ā’1 ļ£¬0ļ£· ,

ļ£C ļ£ø ļ£ 0ļ£ø

D 0

where M is a 4 Ć— 4 matrix to be given explicitly.

Exercise K.280. [12.4, M] Consider the equation for a damped harmonic

oscillator

y (t) + 2Ī²y (t) + Ļ 2 y(t) = f (t),

where Ī² and Ļ are strictly positive real numbers and y : [0, ā) ā’ R satisļ¬es

y(0) = y (0) = 0. Use Greenā™s function methods to show that, if Ī² > Ļ,

t

ā’1

f (s)eā’Ī²(tā’s) sinh(Ī±(t ā’ s)) ds

y(t) = Ī±

0

where Ī± is the positive square root of Ī² 2 ā’ Ļ 2 and obtain a similar result in

the cases when Ļ > Ī² and Ļ = Ī².

It is known that the driving force f is non-zero only when t is very small.

Sketch the behaviour of y(t) for large t and determine the value of Ī² which

causes y to die away as fast as possible. (Interpreting this last phrase in a

reasonable manner is part of your task.)

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Exercise K.281. [13.1, T,] (Although this sequence of exercises seems to

ļ¬nd a natural place here they could have been placed earlier and linked with

Section 11.1.) Let (U, U ) be a complete normed vector space. By Exer-

cise 11.1.15, the space L(U, U ) of continuous linear maps with the operator

norm is complete.

(i) If T ā L(U, U ) and T < 1, show that the sequence Sn = n T j j=0

is Cauchy. Deduce that Sn converges in the operator norm to a limit S =

ā ā’1

and I ā’ S ā¤ T (1 ā’ T )ā’1 .

j

j=0 T . Show also that S ā¤ (1 ā’ T )

(ii) We continue with the notation and assumptions of (i). By looking at

Sn (I ā’ T ), show that S(I ā’ T ) = I. Show also that (I ā’ T )S = I. Conclude

that, if A ā L(U, U ) and I ā’ A < 1, then A is invertible and

Aā’1 ā¤ (1 ā’ I ā’ A )ā’1 and I ā’ Aā’1 ā¤ I ā’ A (1 ā’ I ā’ A )ā’1 .

(iii) Suppose that B ā L(U, U ) is invertible and Bā’C < B ā’1 ā’1 . If we

set A = B ā’1 C, show that A = B ā’1 C is invertible. Show that Aā’1 B ā’1 C = I

and CAā’1 B ā’1 = CBB ā’1 Aā’1 B ā’1 = I, and so C is invertible with inverse

Aā’1 B ā’1 . Show, further that,

C ā’1 ā¤ B ā’1 (1 ā’ B ā’1 B ā’ C ) and

B ā’1 ā’ C ā’1 ā¤ B ā’1 B ā’ C (1 ā’ B ā’1 B ā’ C ).

(iv) Let E be the set of invertible C ā L(U, U ). Show that E is open in

). Show further that, if we deļ¬ne Ī˜ : E ā’ E by Ī˜(C) = C ā’1 ,

(L(U, U ),

then Ī˜ is a continuous function.

(v) Returning to the discussion in (i) and (ii) show that if T < 1 then

(I ā’ T )ā’1 ā’ I ā’ T ā¤ T (1 ā’ T )ā’1

2

and

(I + T )ā’1 ā’ I + T ā¤ T (1 ā’ T )ā’1 .

2

Conclude that

Ī˜(I + T ) = I ā’ T + (T ) T ,

where : L(U, U ) ā’ L(U, U ) is such that (T ) ā’ 0 as T ā’ 0. Con-

clude that Ī˜ is diļ¬erentiable at I. [If you wish to conļ¬ne yourself to ļ¬nite

dimensional spaces, take U ļ¬nite dimensional, but there is no need.]

(vi) Show that Ī˜ is everywhere diļ¬erentiable on E with DĪ˜(B) = Ī¦B ,

where Ī¦B : L(U, U ) ā’ L(U, U ) is the linear map given by

Ī¦B (S) = ā’B ā’1 SB ā’1 .

Check that this reduces to a known result when dim U = 1.

588 A COMPANION TO ANALYSIS

Exercise K.282. (Spectral radius.) [13.1, T, ā‘ ] We continue with the

ideas and notation of Exercise K.281.

(i) Suppose that A ā L(U, U ). Show that

Ajk+r ā¤ Aj k

A r.

= inf nā„1 An 1/n

(ii) Continuing with the hypotheses of (i), show that

exists and, by using the result of (i), or otherwise that

An 1/n

ā’ .

We call the spectral radius of A and write Ļ(A) = .

(iii) Show that ā An converges in the operator norm if Ļ(A) < 1 and

n=0

diverges if Ļ(A) > 1.

(iv) If Ļ(A) < 1 show that I ā’ A is invertible and (I ā’ A)ā’1 = ā An .

n=0

Exercise K.283. [13.1, T] We continue with the ideas of Exercise K.282.

(i) If Ī± ā L(U, U ) and Ī» is a scalar show that Ļ(Ī»Ī±) = |Ī»|Ļ(Ī±).

(ii) Write down a linear map Ī² : Cm ā’ Cm such that Ī² mā’1 = 0 but

Ī² m = 0. What is the value of Ļ(Ī²)?

(iii) If the linear map Ī± : Cm ā’ Cm has m distinct eigenvalues explain

brieļ¬‚y why we can ļ¬nd an invertible linear map Īø : Cm ā’ Cm such that ĪøĪ±Īø ā’1

has a diagonal matrix D with respect to the standard basis. By considering

the behaviour of D n , or otherwise, show that

Ļ(Ī±) = max{|Ī»| : Ī» an eigenvalue of Ī±}.

(For an improvement of this result see Exercise K.285.)

(iv) Give two linear maps Ī±, Ī² : C2 ā’ C2 such that Ļ(Ī±) = Ļ(Ī²) = 0 but

Ļ(Ī± + Ī²) = 1.

Exercise K.284. (Cayley-Hamilton.) This question requires you to know

that any linear map Ī± : Cm ā’ Cm has an upper triangular matrix with re-

spect to some basis. Recall also that, if Ī± : Cm ā’ Cm we can deļ¬ne its

characteristic polynomial ĻĪ± by the condition det(tĪ¹ ā’ Ī±) = ĻĪ± (t) for all

t ā C. (Here and elsewhere, Ī¹ : Cm ā’ Cm is the identity map.)

(i) We work with the standard basis on Cm . Explain why, if Ī± : Cm ā’ Cm

is linear, we can ļ¬nd an invertible linear map Īø : Cm ā’ Cm such that

Ī² = ĪøĪ±Īøā’1 has an upper triangular matrix with respect to the standard

basis.

(ii) Show that, if Ī² : Cm ā’ Cm has an upper triangular matrix with

respect to the standard basis and > 0, then we can ļ¬nd Ī³ : Cm ā’ Cm with

m distinct eigenvalues such that Ī³ ā’ Ī² < .

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(iii) Deduce that, if Ī± : Cm ā’ Cm is linear we can ļ¬nd linear Ī±k : Cm ā’

Cm with m distinct eigenvalues and

Ī±k ā’ Ī± ā’ 0 as k ā’ ā.

(iv) (Pure algebra) If Ī± : Cm ā’ Cm has n distinct eigenvalues, show, by

considering the matrix of Ī± with respect to a basis of eigenvectors that

ĻĪ± (Ī±) = 0.

(v) Show carefully, using (iii) and limiting arguments, that, if Ī± : C m ā’

Cm is linear, then

ĻĪ± (Ī±) = 0.

[This argument is substantially longer than that used in algebra courses and

is repugnant to the soul of any pure minded algebraist. But it has its points

of interest.]

Exercise K.285. [13.1, T, ā‘] We continue with the ideas of Exercise K.283.

(i) If Ī±, Ī² ā L(U, U ) and Ī± and Ī² commute, show that

Ļ(Ī± + Ī²) ā¤ Ļ(Ī±) + Ļ(Ī²).

(Compare Exercise K.283 (iv).)

(ii) By using the ideas of Exercise K.284, or otherwise, show that, if

Ī± : Cm ā’ Cm is linear and > 0, we can ļ¬nd a Ī² : Cm ā’ Cm such that

Ī² < , Ī± and Ī² commute and Ī± + Ī² has m distinct eigenvalues.

(iii) By using (ii), or otherwise, show that, if Ī± : Cm ā’ Cm is linear, then

Ļ(Ī±) = max{|Ī»| : Ī» an eigenvalue of Ī±}.

[You may wish to read or reread Exercises K.99 to K.101 at this point.]

Exercise K.286. [13.1, T, ā‘] In previous questions we have shown that, if

Ī± : Cm ā’ Cm is linear, and we write

Ļ(Ī±) = max{|Ī»| : Ī» an eigenvalue of Ī±}.

then Ī±n 1/n ā’ Ļ(Ī±).

Suppose that b ā Cm , we choose an arbitrary x0 ā Cm and we consider

the sequence

xn+1 = b + Ī±xn .

590 A COMPANION TO ANALYSIS

If Ļ(Ī±) < 1, give two proofs that

xn ā’ (Ī¹ ā’ Ī±)ā’1 b ā’ 0 as n ā’ ā.

(i) By ļ¬nding xn explicitly and using Exercise K.282.

(ii) By using the ļ¬rst paragraph of Exercise K.261.

By considering an eigenvector corresponding to an eigenvalue of largest

modulus show that the sequence will diverge if Ļ(Ī±) > 1 and we choose x0

suitably. Show that, if Ļ(Ī±) = 1, either we can ļ¬nd an x0 such the sequence

diverges or, if the sequence always converges, we can ļ¬nd two starting points

with diļ¬erent limits.

Exercise K.287. [13.1, T, ā‘ ] Consider the problem of solving the equation

Ax = b

where A is an m Ć— m matrix and x and b are column vectors of length

m. If m is small, then standard computational methods will work and, if

m is large and A is a general matrix we have no choice but to use standard

methods. These involve storing all m2 coeļ¬cients and, in the case of Gaussian

elimination require of the order of m3 operations.

Suppose we have to deal with a matrix A such that A is close to I, in

some sense to be determined later in the question, and there are only of

the order of n non-zero coeļ¬cients in I ā’ A in a well organized pattern.

(Such problems arise in the numerical solution of important partial diļ¬eren-

tial equations.) The following method can then be employed. Choose x0 and

deļ¬ne a sequence

xn+1 = b + (I ā’ A)xn .

Using the ideas of earlier exercises show that, under certain conditions, to

be stated, xn will tend to a unique solution of Ax = b. Discuss the rapidity

of convergence, and show that, under certain conditions to be stated, only a

few iterations will be required to get the answer to any reasonable degree of

accuracy. Since each iteration requires, at worst, of the order of m2 opera-

tions, and in many cases only of the order of m operations, this method is

much more eļ¬cient.

The rest of the question consists of elaboration of this idea. We require

Exercise K.286. Suppose that A is an m Ć— m matrix and A = D ā’ U ā’ L

where L is strictly lower triangular, U is strictly upper triangular and D is

diagonal with all diagonal terms non-zero. We seek to solve Ax = b. The

following iterative schemes have been proposed

Jacobi xn+1 = Dā’1 (b + (U + L)xn ),

Gauss-Seidel xn+1 = (D ā’ L)ā’1 (b + U xn ).

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For each of these two schemes give a necessary and suļ¬cient condition in

terms of the spectral radius of an appropriate matrix for the method to

work.

Another iterative scheme uses

xn+1 = (D ā’ ĻL)ā’1 (Ļb + ((1 ā’ Ļ)D + ĻU )xn ),

where Ļ is some ļ¬xed real number. Give a necessary and suļ¬cient condition,

in the form Ļ(H) < 1 where H is an appropriate matrix, for the method to

work. By showing that

det H = (1 ā’ Ļ)m ,

or otherwise, show that the scheme must fail if Ļ < 0 or Ļ > 2.

Exercise K.288. [13.1, S, ā‘ā‘ ] (This is a short question, but requires part (iv)

of Exercise K.281.) Show that Theorem 13.1.13 can be strengthened by

adding the following sentence at the end. ā˜Moreover Df |ā’1 is continuous on

B

B.ā™

Exercise K.289. [13.1, T, ā‘ā‘ ] This is another exercise in the ideas of Ex-

ercise K.281. We work in (U, U ) as before.

(i) Show that, if Ī± ā L(U, U ), we can ļ¬nd exp Ī± ā L(U, U ) such that

n

Ī±r

ā’ exp Ī± ā’ 0

r!

r=0

as n ā’ ā.

(ii) Show carefully that, if Ī± and Ī² commute,

exp Ī± exp Ī² = exp(Ī± + Ī²).

(iii) Show that if Ī± and Ī² are general (not necessarily commuting) ele-

ments of L(U, U ), then

hā’2 (exp(hĪ±) exp(hĪ²) ā’ exp(hĪ²) exp(hĪ±)) ā’ (Ī±Ī² ā’ Ī²Ī±) ā’ 0

as the real number h ā’ 0.

Conclude that, in general, exp(Ī±) exp(Ī²) and exp(Ī²) exp(Ī±) need not be

equal. Deduce also that exp(Ī± + Ī²) and exp(Ī±) exp(Ī²) need not be equal.

(iv) Show carefully (you must bear part (iii) in mind) that exp : L(U, U ) ā’

L(U, U ) is everywhere continuous.

592 A COMPANION TO ANALYSIS

Exercise K.290. [13.1, P, ā‘ ] (i) Consider the map Ī˜3 : L(U, U ) ā’ L(U, U )

given by Ī˜3 (Ī±) = Ī±3 . Show that Ī˜ is everywhere diļ¬erentiable with

DĪ˜3 (Ī±)Ī² = Ī²Ī±2 + Ī±Ī²Ī±2 + Ī±2 Ī².

(ii) State and prove the appropriate generalisation to the map Ī± ā’ Ī± m

with m a positive integer.

(iii) Show that exp, deļ¬ned in Exercise K.289, is everywhere diļ¬erentiable.

[This requires care.]

Exercise K.291. [13.1, P] Suppose U is a ļ¬nite dimensional vector space

over C. Let Ī± : U ā’ U be a linear map. If Ī± has matrix representation A

with respect to some basis, explain why, as N ā’ ā, the entries of the matrix

N n

n=0 A /n! converge to the entries of a matrix exp A which represents exp Ī±

with respect to the given basis. (See Exercise K.289.)

It is a theorem that any Ī± ā L(U, U ) has an upper triangular matrix with

respect to some basis. By using this fact, or otherwise, show that

det(exp Ī±) = eTrace Ī± .

Exercise K.292. [13.1, P] We work in the space M2 (R) of 2 Ć— 2 real ma-

trices. We give M2 (R) the associated operator norm.

(i) Show that the map S : M2 (R) ā’ M2 (R) given by S(A) = A2 is

everywhere diļ¬erentiable with DS(A)B = AB + BA. (If you have done

Exercise K.290, you may just quote it.)

(ii) Show that the matrix equation

01

A2 =

00

has no solution.

(iii) Calculate explicitly all the solutions of

2

ab

= I.

cd

Describe geometrically the linear maps associated with the matrices A such

that A2 = I and det A = ā’1. Describe geometrically the linear maps associ-

ated with the matrices A such that A2 = I and det A = 1. Describe geomet-

rically the linear maps associated with the matrices A such that A2 = I and

A is diagonal.

(iv) Show that there are open sets U and V containing 0 (the zero matrix)

such that the equation

A2 = I + X

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has exactly one solution of the form A = I + Y with Y ā V for each X ā U .

(v) Show that we can not ļ¬nd open sets U and V containing 0 (the zero

matrix) such that the equation

A2 = I + X

has exactly one solution of the form

10

A= +Y

0 ā’1

with Y ā V for each X ā U . Identify which hypothesis of the inverse function

theorem (Theorem 13.1.13) fails to hold and show, by direct calculation, that

is does indeed fail.

(vi) For which B is it true that B 2 = I and we can ļ¬nd open sets U and

V containing 0 (the zero matrix) such that the equation

A2 = I + X

has exactly one solution of the form A = B + Y with Y ā V for each X ā U .

Give reasons for your answer.

Exercise K.293. [13.1, P, G] This question requires some knowledge of

eigenvectors of symmetric linear maps. We work in R3 with the usual inner

product. Suppose Ī± : R3 ā’ R3 is an antisymmetric linear map (that is

Ī±T = ā’Ī±). Show that x and Ī±x are orthogonal and that the eigenvalues of

Ī±2 must be non-positive real numbers. By considering eigenvectors of Ī± and

Ī±2 show that we can always ļ¬nd Āµ ā R and three orthonormal vectors e1 , e2

and e3 , such that

Ī±e1 = Āµe2 , Ī±e2 = ā’Āµe1 , Ī±e3 = 0.

By choosing appropriate axes and using matrix representations show that

exp Ī± is a rotation.

Exercise K.294. [13.1, P, G] Show that every linear map Ī± : Rm ā’ Rm

is the sum of a symmetric and an antisymmetric linear map. Suppose Ī± is

an orthogonal map (that is, Ī±Ī±T = Ī¹) with Ī¹ ā’ Ī± < with very small.

Show that

2

Ī±=Ī¹+ Ī²+ Ī³

with Ī² , Ī³ ā¤ 2.

594 A COMPANION TO ANALYSIS

Exercise K.295. [13.3, M] (Treat this as a ā˜methods questionā™.) The four

vertices A, B, C, D of a quadrilateral lie in anti-clockwise order on a circle

radius a and center O. We write 2Īø1 = ā AOB, 2Īø2 = ā BOC, 2Īø3 = ā COD,

2Īø4 = ā DOA. Find the area of the quadrilateral and state the relation that

Īø1 , Īø2 , Īø3 and Īø4 must satisfy.

Use Lagrangeā™s method to ļ¬nd the form of a quadrilateral of greatest area

inscribed in a circle of radius a. (Treat this as a ā˜methods questionā™.)

Use Lagrangeā™s method to ļ¬nd the form of an n-gon of greatest area

inscribed in a circle [n ā„ 3].

Use Lagrangeā™s method to ļ¬nd the form of an n-gon of least area circum-

scribing a circle [n ā„ 3].

[Compare Exercise K.40.]

Exercise K.296. [13.3, T] Let p and q be strictly positive real numbers

with pā’1 + q ā’1 = 1. Suppose that y1 , y2 , . . . , yn , c > 0. Explain why there

must exist x1 , x2 , . . . , xn ā„ 0 with n xp = c and

j=1 j

n n n

tp = c.

xj yj ā„ tj yj whenever t1 , t2 , . . . , tn ā„ 0 with j

j=1 j=1 j=1

Use the Lagrange multiplier method to ļ¬nd the xj . Deduce from your

answer that

1/p 1/q

n n n

|aj |p |bj |q

|aj bj | ā¤

j=1 j=1 j=1

whenever aj , bj ā C. Under what conditions does equality hold?

This gives an alternative proof of the ļ¬rst result in Exercise K.191 (i).

Exercise K.297. (The parallelogram law.) [14.1, T] Except in the last

part of this question we work in a real normed vector space (V, ).

(i) Suppose that V has a real inner product , such that x, x = x 2

for all x ā V . Show that

2 2 2 2

+ xā’y

x+y =2 x +2 y

for all x, y ā V . (This is called the parallelogram law.)

(ii) Show also that

2 2

ā’ xā’y

4 x, y = x + y

for all x, y ā V . (This is called the polarisation identity.)

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(iii) Use the parallelogram law to obtain a relation between the lengths

of the sides and the diagonals of a parallelogram in Euclidean space.

(iv) Prove the inequality | x ā’ y | ā¤ x ā’ y and use it together with

the polarisation identity and the parallelogram law to give another proof of

the Cauchy-Schwarz inequality.

(v) Suppose now that (V, , ) is a complex inner product space with

norm derived from the inner product. Show that the parallelogram law

holds in the same form as before and obtain the new polarisation identity

2 2 2

ā’ i x ā’ iy 2 .

ā’ xā’y

4 x, y = x + y + i x + iy

(vi) Show that the uniform norm on C([0, 1]) is not derived from an inner

product. (That is to say, there dooes not exist an inner product , with

f 2 = langlef, f for all f ā C([0, 1]).

ā

Exercise K.298. [14.1, T, ā‘ ] The parallelogram law of Question K.297

actually characterises norms derived from an inner product although the

proof is slightly trickier than one might expect.

(i) Let (V, ) be real normed space such that

2 2 2 2

+ xā’y

x+y =2 x +2 y

for all x, y ā V . It is natural to try setting

x, y = 4ā’1 2 2

ā’ xā’y

x+y .

Show that x, y = y, x for all x, y ā V and that x, x = x 2 (so that,

automatically, x, x ā„ 0 with equality if and only if x = 0).

(ii) The remaining inner product rules are harder to prove. Show that

2 2 2 2

+ u+vā’w

u+v+w =2 u+v +2 w

and use this to establish that

u + w, v + u ā’ w, v = 2 u, v (1)

for all u, v, w ā V . Use equation (1) to establish that

2u, v = 2 u, v (2)

and then use equations (1) and (2) to show that

x, v + y, v = x + y, v

596 A COMPANION TO ANALYSIS

for all x, y, v ā V .

(iii) Establish the equation

Ī»x, y = Ī» x, y

for all positive integer values of Ī», then for all integer values, for all rational

values and then for all real values of Ī».

(iv) Use the fact that the parallelogram law characterises norms derived

from an inner product to give an alternative proof of Lemma 14.1.11 in the

real case.

(v) Extend the results of this question to complex vector spaces.

Exercise K.299. [14.1, P] Suppose (X, d) is a complete metric space with

a dense subset E. Suppose that E is a vector space (over F where F = R

or F = C) with norm E such that d(x, y) = x ā’E y E for all x, y ā

E. Suppose further that there is map ME : E 2 ā’ E such that, writing

ME (x, y) = xy, we have

(i) x(yz) = (xy)z,

(ii) (x + y)z = xz + yz, z(x + y) = zx + zy

(iii) (Ī»x)y = Ī»(xy), x(Ī»y) = Ī»(xy),

(iv) xy ā¤ x y ,

for all x, y, z ā E and Ī» ā F. Show that, if X is given the structure of a

normed vector space as in Lemma 14.1.9, then we can ļ¬nd a map M : X 2 ā’

X such that M (x, y) = ME (x, y) for all x, y ā E and M has properties (i)

to (iv) (with E replaced by X). Show that, if ME (x, y) = ME (y, x) for all

x, y ā E, then M (x, y) = M (y, x) for all x, y ā X. Show that, if there

exists an e ā E such that ME (x, e) = ME (e, x) = x for all x ā E, then

M (x, e) = M (e, x) = x for all x ā X.

Exercise K.300. [14.1, T] This neat proof that every metric space (E, d)

can be completed is due to Kuratowski14 .

(i) Choose e0 ā E. For each e ā E deļ¬ne fe (t) = d(e, t) ā’ d(e0 , t) [t ā E].

Show that fe ā C(E), where C(E) is the space of bounded continuous function

g : E ā’ R.

(ii) Give C(E) the usual uniform norm. Show that

fu ā’ fv = d(u, v)

for all u, v ā E.

(iii) Let Y be the closure of {fe : e ā E}. By using Theorem 11.3.7, or

Ė

otherwise show that Y with the metric d inherited from C(E) is complete.

Show that (E, d) has a completion by considering the map Īø : E ā’ Y given

by Īø(e) = fe .

14

I take it from from a book [15] crammed with neat proofs.

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Exercise K.301. [14.1, P] Results like Lemma 14.1.9 rely on a strong link

between the algebraic operation and the metric. From one point of view this

question consists of simple results dressed up in jargon but I think they shed

some light on the matter.

(i) (This just sets up a bit of notation.) Suppose that (X, d) is a metric

space. Show that d2 : E 2 ā’ R given by

d2 ((x, y), (x , y )) = d(x, x ) + d(y, y )

deļ¬nes a metric on X 2 . Show that, if (X, d) is complete, so is (X 2 , d2 ).

(ii) Consider X = [0, ā) with the usual Euclidean metric d and E =

(0, ā). Show that (X, d) is complete, E is a dense subset of X and that,

if we write ME (x, y) = xy (that is if M is ordinary multiplication), then

(E, M ) is a group and ME : (E 2 , d2 ) ā’ (E, d) is continuous. Show, however,

that there does not exist a continuous map M : (X 2 , d2 ) ā’ (X, d) with

M (x, y) = ME (x, y) for all x, y ā E such that (X, M ) is a group.

(iii) Consider X = [0, ā) with the usual Euclidean metric d and E =

(0, ā) ā© Q. Show that (X, d) is complete, E is a dense subset of X and that,

if we write ME (x, y) = xy, then (E, M ) is a group and ME : (E 2 , d2 ) ā’ (E, d)

is continuous. Show, however, that there does not exist a continuous map

M : (X 2 , d2 ) ā’ (X, d) with M (x, y) = ME (x, y) for all x, y ā E such that

(X, M ) is a group.

(iv) Consider X = (0, ā). Show that if we write

d(x, y) = | log x ā’ log y|

then (X, d) is a metric space. Let E = (0, ā) ā© Q. Show that (X, d) is

complete, E is a dense subset of X and that, if we write ME (x, y) = xy, then

(E, M ) is a group and ME : (E 2 , d2 ) ā’ (E, d) is continuous. Show that there

exists a continuous map M : (X 2 , d2 ) ā’ (X, d) with M (x, y) = ME (x, y) for

all x, y ā E such that (X, M ) is a group.

Exercise K.302. [14.1, P] (i) Observe that R is a group under addition. If

E is a subgroup of R which is also a closed set with respect to the Euclidean

metric, show that either E = R or

E = {nĪ± : n ā Z}

for some Ī± ā R.

(ii) Observe that

S 1 = {Ī» ā C : |Ī»| = 1}

598 A COMPANION TO ANALYSIS

is a group under multiplication. What can you say about subgroups E of S 1

which are closed with respect to the usual metric?

(iii) Observe that Rm is a group under vector addition. What can you

say about subgroups E of Rm which are closed with respect to the Euclidean

metric?

Exercise K.303. [14.1, T, ā‘ā‘ ] Exercise K.56 is not important in itself, but

the method of its proof is. Extend the result and proof to f : E ā’ R where

E is a dense subspace of a metric space (X, d).

Can the result be extended to f : E ā’ Y where E is a dense subspace

of a metric space (X, d) and (Y, Ļ) is a metric space? (Give a proof or

counterexample.) If not, what natural extra condition can we place on (Y, Ļ)

so that the result can be extended?

Exercise K.304. [14.1, P, S, ā‘ ] Suppose that f : R ā’ R satisļ¬es

|f (x) ā’ f (y)| ā¤ (x ā’ y)2 for all x, y ā R.

Show that f is constant.

Show that the result remains true if we replace R by Q. Explain why this

is consistent with examples of the type given in Example 1.1.3

Exercise K.305. [14.1, P] Let (X, d) be a metric space with the Bolzano-

Weierstrass property. Show that, given any > 0, we can ļ¬nd a ļ¬nite set of

points x1 , x2 , . . . xn such that the open balls B(xj , ) centre xj and radius

cover X (that is to say, n B(xj , ) = X). (This result occurs elsewhere

j=1

both in the main text and exercises but it will do no harm to reprove it.)

Deduce that (X, d) has a countable dense subset.

Give an example of a complete metric space which does not have the

Bolzano-Weierstrass property but does have a countable dense subset. Give

an example of a metric space which is not complete but does have a countable

dense subset. Give an example of a complete metric space which does not

have a countable dense subset.

Exercise K.306. [14.1, P] (i) By observing that every open interval con-

tains a rational number, or otherwise, show that every open subset of R

(with the standard Euclidean metric) can be written as the countable union

of open intervals.

(ii) Let (X, d) be a metric space with a countable dense subspace. Show

that every open subset of X can be written as the countable union of open

balls.

599

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

(iii) Consider R2 . Deļ¬ne Ļ : R2 Ć— R2 ā’ R by

1 if y1 = y2

Ļ((x1 , y1 ), (x2 , y2 )) =

min(1, |x1 ā’ x2 |) if y1 = y2 .

Show that Ļ is a metric and that Ļ is complete. Show that, if we work in

(R2 , Ļ),

V = {(x, y) : |x| < 1, y ā R}

is an open set that can not be written as the countable union of open balls.

(iv) Let (X, d) be a metric space with a countable dense subspace. Show

that every open ball can be written as the countable union of closed balls.

Show that every open set can be written as the countable union of closed

balls. Show that, if U is an open set, we can ļ¬nd bounded closed sets Kj with

Kj+1 ā Kj [1 ā¤ j] and ā Kj = U . [This result is useful for spaces like Rn

j=1

with the usual metric where we know, in addition, that bounded closed sets

have the Bolzano-Weierstrass property.]

Exercise K.307. [14.1, P] The previous question K.306 dealt with general

metric spaces. Apart from the last part this question deals with the particular

space R with the usual metric. We need the notion of an equivalence relation.

(i) Let U be an open subset of R. If x, y ā U , write x ā¼ y if there is an

open interval (a, b) ā‚ U with x, y ā (a, b). Show that ā¼ is an equivalence

relation on U .

(ii) Write [x] = {y ā U : y ā¼ x} for the equivalence class of some x ā U .

If [x] is bounded, show, by considering the inļ¬mum and supremum of [x], or

otherwise, that [x] is an open interval. What can we say about [x] if it is

bounded below but not above? Prove your answer carefully. What can we

say about [x] if it is bounded above but not below? What can we say if [x]

is neither bounded above nor below?

(iii) Show that U is the disjoint union of a collection C of sets of the form

(a, b), (c, ā), (ā’ā, c) and R.

(iv) Suppose that U is the disjoint union of a collection C of sets of the

form (a, b), (c, ā), (ā’ā, c) and R. If J ā C explain why there exists an

I ā C with I ā J. Explain why, if a is an end point of J which is not an end

point of I, there must exist a J ā C with a ā J . Hence, or otherwise, show

that J = I. Conclude that C = C.

(v) Show that C is countable.

(vi) We saw in part (iv) that C is uniquely deļ¬ned and this raises the

possibility of deļ¬ning the ā˜lengthā™ of U to be the sum of the lengths of the

600 A COMPANION TO ANALYSIS

intervals making up C. However, this approach fails in higher dimensions.

The rest of this question concerns R2 with the usual metric.

Show that the open square (ā’a, a) Ć— (ā’a, a) is not the union of disjoint

open discs. [It may be helpful to look at points on the boundary of a disc

forming part of such a putative union.]

Show that the open disc {(x, y) : x2 + y 2 < 1} is not the union of disjoint

open squares.

Exercise K.308. [14.1, T] We say that metric spaces (X, d) and (Y, Ļ) are

homeomorphic if there exists a bijective map f : X ā’ Y such that f and

f ā’1 are continuous. We say that f is a homeomorphism between X and Y .

(i) Show that homeomorphism is an equivalence relation on metric spaces.

(ii) If f : X ā’ Y is a homeomorphism, show that U is open in (X, d) if

and only if f (U ) is open in (Y, Ļ).

(iii) By constructing an explicit homeomorphism, show that R with the

usual metric is homeomorphic to the open interval (ā’1, 1) with the usual

metric. Deduce that the property of completeness is not preserved under

homeomorphism.

(iv) By constructing an explicit homeomorphism show that I = (ā’1, 1)

with the usual metric is homeomorphic to

J = {z ā C : |z| = 1, z = 1}

with the usual metric. Show that [ā’1, 1] with the usual metric is a completion

of I. Find a completion of J.

Explain brieļ¬‚y why the completion of I adds two points but the comple-

tion of J adds only one.

Exercise K.309. [14.1, T] (i) Suppose (X, d) is a metric space with the

Bolzano-Weierstrass property and (Y, Ļ) is a any metric space. If f : X ā’ Y

is a bijective continuous function show that (Y, Ļ) has the Bolzanoā“Weierstrass

property and that f ā’1 : Y ā’ X is uniformly continuous. (Note that we have

shown that, in the language of Exercise K.308, (X, d) and (Y, Ļ) are homeo-

morphic.)

(ii) Look brieļ¬‚y at Exercise 5.6.8. Which results (if any) of that exercise

can can be obtained using (i)?

(iii) Consider R with the usual metric. Give an example of a uniformly

continuous bijective map f : R ā’ R with f ā’1 not uniformly continuous.

(iv) Let d be the usual metric on R and Ļ the discrete metric on R. Let

f : (R, Ļ) ā’ (R, d) be given by f (x) = x. Show that f is is a bijective

continuous function but f ā’1 is not continuous.

601

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

Exercise K.310. [14.1, T, ā‘ ] Suppose that (X, d) is a metric space with

the Bolzano-Weierstrass property. Explain (by referring to Exercise K.305,

if necessary) why we can ļ¬nd a countable dense subset {x1 , x2 , x3 , . . . },

say, for X. Consider l 2 with its usual norm (see Exercise K.188). Show that

the function f : X ā’ l2 given by

f (x) = (d(x, x1 ), 2ā’1 d(x, x2 ), 2ā’2 d(x, x2 ), . . . )

is well deļ¬ned, continuous and injective. Deduce that f (X) is homeomorphic

to X. [Thus l2 contains a subsets homeomorphic to any given metric space

with the Bolzano-Weierstrass property.]

Exercise K.311. [14.1, P] Suppose that (X, d) and (Y, Ļ) are metric spaces

and f : X ā’ Y is a continuous surjective map.

(i) Suppose that Ļ(f (x), f (x )) ā¤ Kd(x, x ) for all x, x ā X and some

K > 0. If (X, d) is complete, does it follow that (Y, Ļ) is complete? If

(Y, Ļ) is complete, does it follow that (X, d) is complete? Give proofs or

counterexamples as appropriate.

(ii) Suppose that Ļ(f (x), f (x )) ā„ Kd(x, x ) for all x, x ā X and some

K > 0. If (X, d) is complete, does it follow that (Y, Ļ) is complete? If

(Y, Ļ) is complete, does it follow that (X, d) is complete? Give proofs or

counterexamples as appropriate.

Exercise K.312. [14.1, P] Let X be the space of open intervals Ī± = (a, b)

with a < b in R. If Ī±, Ī² ā X, then the symmetric diļ¬erence Ī± Ī² consists

of the empty set, one open interval or two disjoint open intervals. We deļ¬ne

d(Ī±, Ī²) to be the total length of the intervals making up Ī± Ī². Show that d

is a metric on X.

Show that the completion of (X, d) contains precisely one further point.

Exercise K.313. [14.1, P] Consider the set N of non-negative integers. If

n ā N and n = 0, then there exist unique r, s ā N with s an odd integer and

n = s2r . Write Ī»(n) = r. If n, m ā N and n = m we write

d(n, m) = 2ā’Ī»(|nā’m|) .

We take d(n, n) = 0.

(i) Show that d is a metric on N.

(ii) Show that the open ball centre 1 and radius 1 is closed.

(iii) Show that no one point set {n} is open.

(iv) Show that the function f : N ā’ N given by f (x) = x2 is continuous.

(v) Show that the function g : N ā’ N given by f (x) = 2x is not continuous

at any point of N.

602 A COMPANION TO ANALYSIS

(vi) Show that d is not complete. (Be careful. You must show that your

Cauchy sequence does not converge to any point of N.)

Reļ¬‚ect on what the completion might look like. (You are not called to

come to any conclusion.)

Exercise K.314. [Appendix C, P] Show that there exists a continuous

function F : R2 ā’ R which is inļ¬nitely diļ¬erentiable at at every point except

0, which has directional derivative zero in all directions at 0 but which is not

diļ¬erentiable at 0.

Bibliography

[1] F. S. Acton. Numerical Methods That Work. Harper and Row, 1970.

[2] A. F. Beardon. Limits. A New Approach to Real Analysis. Springer,

1997.

[3] R. Beigel. Irrationality without number theory. American Mathematical

Monthly, 98:332ā“335, 1991.

[4] B. Belhoste. Augustin-Louis Cauchy. A biography. Springer, 1991.

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[5] D. Berlinski. A Tour of the Calculus. Pantheon Books, New York, 1995.

[6] P. Billingsley. Probability and Measure. Wiley, 1979.

[7] E. Bishop and D. Bridges. Constructive Analysis. Springer, 1985.

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of Dolciani Mathematical Expositions. MAA, 1995.

[9] J. R. Brown. Philosophy of Mathematics. Routledge and Kegan Paul,

1999.

[10] J. C. Burkill. A First Course in Mathematical Analysis. CUP, 1962.

[11] R. P. Burn. Numbers and Functions. CUP, 1992.

[12] W. S. Churchill. My Early Life. Thornton Butterworth, London, 1930.

[13] J. DieudonnĀ“. Foundations of Modern Analysis. Academic Press, 1960.

e

[14] J. DieudonnĀ“. Inļ¬nitesimal Calculus. Kershaw Publishing Company,

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London, 1973. Translated from the French Calcul Inļ¬nitĀ“simal pub-

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604 A COMPANION TO ANALYSIS

[15] R. M. Dudley. Real Analysis and Probability. Wadsworth and Brooks,

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bers. CUP, 1938.

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C. A. Hedrick.

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London Mathematical Society, 23:557ā“62, 1991.

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Index

abuse of language, 422 Big Oh and little oh, 506

algebraists, dislike metrics, 127, 589 bijective function, 475

alternating series test, 78 binomial expansion

antiderivative, existence and uniqueness, for general exponent, 297

186 positive integral exponent, 562

area, general problems, 169ā“172, 214ā“217, binomial theorem, 297, 562

229ā“231 bisection, bisection search, see lion hunt-

authors, other ing

Beardon, 56, 395 Bishopā™s constructive analysis, 415ā“419

Berlinski, 20 Bolzano-Weierstrass

Billingsly, 230 and compactness, 421

Boas, 60, 152, 211 and total boundedness, 274

Bourbaki, 376, 422 equivalent to fundamental axiom, 40

for R, 38ā“39

Burn, 62

for closed bounded sets in Rm , 49

Conway, 449

DieudonnĀ“, viii, 25, 60, 154, 206

e for metric spaces, 272ā“274

in Rm , 47

Halmos, 375, 391

Hardy, viii, 43, 83, 103, 154, 297 bounded variation, functions of, 181, 516ā“

Klein, 113, 422 519

Kline, 375 brachistochrone problem, 190

Littlewood, 81

Petard, H., 16 calculus of variations

Plato, 28 problems, 198ā“202

PoincarĀ“, 376

e successes, 190ā“198

axiom used, 258

fundamental, 9, 12, 22, 374 Cantor set, 552

of Archimedes, 10, 12, 373, 412 Cauchy

of choice, 172, 252 condensation test, 76

axioms father of modern analysis, viii

for an ordered ļ¬eld, 379 function not given by Taylor series,

143

general discussion, 242, 364ā“365, 375ā“

377 mean value theorem, 457

Zermelo-Fraenkel, 375 proof of binomial theorem, 562

sequence, 67, 263

balls solution of diļ¬erential equations, 563

open and closed, 50, 245 Cauchy-Riemann equations, 479

packing, 233ā“235 Cauchy-Schwarz inequality, 44

packing in Fn , 237ā“238 Cayley-Hamilton theorem, 588

2

Banach, 242, 303 chain rule

Bernstein polynomial, 542 many dimensional, 131ā“132

607

608 A COMPANION TO ANALYSIS

one dimensional, 102ā“104 convergence tests for sums

Chebychev, see Tchebychev Abelā™s, 79

chords, 475 alternating series, 78

closed bounded sets in Rm Cauchy condensation, 76

and Bolzano-Weierstrass, 49 comparison, 70

and continuous functions, 56ā“58, 66 discussion of, 465

compact, 421 integral comparison, 208

nested, 59 ratio, 76

closed sets convergence, pointwise and uniform, 280

complement of open sets, 51, 245 convex

deļ¬nition for Rm , 49 function, 451, 498ā“499

deļ¬nition for metric space, 244 set, 447, 571

key properties, 52, 245 convolution, 565ā“566

closure of a set, 526 countability, 383ā“386

comma notation, 126, 148 critical points, see also maxima and min-

compactness, 421, 536 ima, 154ā“160, 163ā“167, 340

completeness

D notation, 126, 150

deļ¬nition, 263

Darboux, theorems of, 452, 492

proving completeness, 267

decimal expansion, 13

proving incompleteness, 264

delta function, 221, 320

completion

dense sets as skeletons, 12, 356, 543

discussion, 355ā“358

derivative

existence, 362ā“364, 596

complex, 288ā“289

ordered ļ¬elds, 411ā“413

directional, 125

structure carries over, 358ā“361

general discussion, 121ā“127

unique, 356ā“358

in applied mathematics, 152, 401ā“

constant value theorem

404

false for rationals, 2

in many dimensions, 124

many dimensional, 138

in one dimension, 18

true for reals, 20

left and right, 424

construction of

C from R, 367ā“368 more general, 253

Q from Z, 366ā“367 not continuous, 452

R from Q, 369ā“374 partial, 126

Z from N, 366 devilā™s staircase, 552

diagrams, use of, 98

continued fractions, 436ā“440

diļ¬erential equations

continuity, see also uniform continuity

and Greenā™s functions, 318ā“326

discussed, 7, 388ā“391, 417

and power series, 294, 563

of linear maps, 128, 250ā“253

Eulerā™s method, 577ā“580

pointwise, 7, 53, 245

existence and uniqueness of solutions,

via open sets, 54, 246

305ā“318

continuous functions

diļ¬erentiation

exotic, 2, 549ā“554

Fourier series, 302

integration of, 182ā“186

on closed bounded sets in Rm , 56ā“59 power series, 291, 557ā“558

term by term, 291

continuum, models for, see also reals and

under the integral

rationals, 25ā“28, 418ā“419

ļ¬nite range, 191

contraction mapping, 303ā“305, 307, 330,

408 inļ¬nite range, 287

609

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

Diniā™s theorem, 542 Greek rigour, 29, 365, 376, 521

directed set, 396 Greenā™s functions, 318ā“326, 583ā“586

dissection, 172

Hahn-Banach for Rn , 447

dominated convergence

Hausdorļ¬ metric, 534

for some integrals, 547

for sums, 84 Heine-Borel theorem, 449

duck, tests for, 369 Hessian, 157

hill and dale theorem, 164ā“166

economics, fundamental problem of, 58 HĀØlderā™s inequality, 531, 533

o

escape to inļ¬nity, 84, 283ā“284 homeomorphism, 600

Euclidean homogeneous function, 480

geometry, 364ā“365

norm, 44 implicit function theorem

Euler discussion, 339ā“347

method for diļ¬erential equations, 577ā“ statement and proof, 343ā“344

580 indices, see powers

on homogeneous functions, 480 inequality

Eulerā™s Ī³, 467 arithmetic-geometric, 451

Euler-Lagrange equation, 194 Cauchy-Schwarz, 44

exponential function, 91ā“98, 143, 317, 417, HĀØlderā™s, 531, 533

o

497, 591 Jensenā™s, 450, 498

extreme points, 447ā“448 Ptolomeyā™s, 443

reverse HĀØlder, 532

o

Father Christmas, 172

Tchebychev, 222

ļ¬xed point theorems, 17, 303ā“304

inļ¬mum, 34

Fourier series, 298ā“302

inļ¬nite

Fubiniā™s theorem

products, 472, 561

for inļ¬nite integrals, 512

sums, see sums

for integrals of cts fns, 213, 510

injective function, 475

for sums, 90

inner product

full rank, 345

completion, 360, 364

functional equations, 477ā“479

for l2 , 531

fundamental axiom, 9

for Rn , 43

fundamental theorem of algebra

integral kernel, example of, 326

proof, 114ā“117

integral mean value theorem, 490

statement, 113

integrals

theorem of analysis, 114, 120

along curves, 228ā“229, 231

fundamental theorem of the calculus

and uniform convergence, 282

discussion of extensions, 186

improper (or inļ¬nite), 207ā“211

in one dimension, 184ā“186

of continuous functions, 182ā“186

over area, 212ā“217

Gabrielā™s horn, 521

principle value, 211

Gaussian quadrature, 544

Riemann, deļ¬nition, 172ā“174

general principle

Riemann, problems, 205ā“206, 214

of convergence, 68, 263, 412

Riemann, properties, 174ā“181

of uniform convergence, 280

Riemann-Stieltjes, 217ā“224, 519

generic, 164

Riemann-Stieltjes, problems, 220

geodesics, 254ā“260

vector-valued, 202ā“204

global and local, contrasted, 65, 123, 142ā“

144, 155, 160, 164, 314ā“317, 341 integration

610 A COMPANION TO ANALYSIS

by parts, 189 method, 350ā“351

by substitution, 187 necessity, 350

numerical, 495ā“497, 544 suļ¬ciency, 353

Riemann versus Lebesgue, 206ā“207 Leader, examples, 528

term by term, 287 left and right derivative, 424

interchange of limits Legendre polynomials, 544ā“545, 559

derivative and inļ¬nite integral, 287 Leibniz rule, 580

derivative and integral, 191 limits

general discussion, 83ā“84 general view of, 395ā“400

inļ¬nite integrals, 512 in metric spaces, 243

integral and sum, 287 in normed spaces, 244

integrals, 213 more general than sequences, 55ā“56

limit and derivative, 285, 286 pointwise, 280

sequences in Rm , 46

limit and integral, 282ā“284

limit and sum, 84 sequences in ordered ļ¬elds, 3ā“7

partial derivatives, 149 uniform, 280

sums, 90 limsup and liminf, 39

interior of a set, 526 lion hunting

intermediate value theorem in C, 42

equivalent to fundamental axiom, 22 in R, 15ā“16, 58, 491ā“492

false for rationals, 2 in Rm , 48

not available in constructive analy- Lipschitz

sis, 418 condition, 307

obvious?, 25ā“28 equivalence, 248

true for reals, 15 logarithm

international date line, 108 for (0, ā), 104ā“106, 476, 497

inverse function theorem non-existence for C \ {0}, 108ā“109,

alternative approach, 407ā“410 315ā“317

gives implicit function theorem, 342 what preceded, 475

many dimensional, 337

one dimensional, 106, 402 Markov chains, 571ā“574

inverses in L(U, U ), 336, 587 maxima and minima, 58, 154ā“160, 194ā“

irrationality of 202, 347ā“354

e, 97 Maxwell

ā 467

Ī³?, hill and dale theorem, 164

2, 432 prefers coordinate free methods, 46,

irrelevant m, 269 121

isolated points, 263 mean value inequality

for complex diļ¬erentiation, 289

Jacobian

for reals, 18ā“20, 22, 36, 60

determinant, 406

many dimensional, 136ā“138

matrix, 127

mean value theorem

Jensenā™s inequality, 450, 498

Cauchyā™s, 457

discussion of, 60

Kant, 28, 364

fails in higher dimensions, 139

kindness to animals, 514

for higher derivatives, 455

Krein-Milman for Rn , 448

for integrals, 490

statement and proof, 60

Lagrangian

limitations, 353 metric

611

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

as measure of similarity, 278ā“279 partial derivatives

British railway non-stop, 243 and Jacobian matrix, 127

British railway stopping, 243 and possible diļ¬erentiability, 147, 161

complete, 263 deļ¬nition, 126

completion, 363 notation, 126, 148, 150, 401ā“404, 423

deļ¬nition, 242 symmetry of second, 149, 162

derived from norm, 242 partition, see dissection

discrete, 273 pass the parcel, 339

Hausdorļ¬, 534 piecewise deļ¬nitions, 425

Lipschitz equivalent, 248 placeholder, 241, 350, 422

totally bounded, 273 pointwise compared with uniform, 65, 280,

MĀØbius transformation, 255ā“261

o 282

monotone convergence power series

for sums, 470 addition, 459

and diļ¬erential equations, 294, 563

neighbourhood, 50, 245 composition, 470

non-Euclidean geometry, 364ā“365 convergence, 71

norm diļ¬erentiation, 291, 557ā“558

all equivalent on Rn , 248 limitations, 143, 298

completion, 358, 364 many variable, 469

deļ¬nition, 241 multiplication, 94

Euclidean, 44 on circle of convergence, 71, 80

operator, 128, 253, 481 real, 293

sup, 276 uniform convergence, 290

uniform, 275, 277

uniqueness, 293

notation, see also spaces

powers

Dij g, 150

beat polynomials, 434

Dj g, 126

deļ¬nition of, 109ā“113, 294ā“296, 555ā“

and Ā· , 241

557

Ī¹, 588

primary schools, Hungarian, 384

x, y , 43

primes, inļ¬nitely many, Eulerā™s proof, 473

g,ij , 148

probability theory, 221ā“224, 240ā“241

g,j , 126

Ptolomeyā™s inequality, 443

z ā— , 119

x Ā· y, 43

quantum mechanics, 27

non-uniform, 422ā“423

nowhere diļ¬erentiable continuous func-

radical reconstructions of analysis, 375,

tion, 549

415ā“419

radius of convergence, see also power se-

open problems, 80, 468

ries, 71, 78, 290, 460

open sets

rationals

can be closed, 273

countable, 385

complement of closed sets, 51, 245

dense in reals, 12

deļ¬nition for Rm , 50

not good for analysis, 1ā“3

deļ¬nition for metric space, 244

reals, see also continuum, models for

key properties, 51, 245

and fundamental axiom, 9

operator norm, 128, 253, 481

existence, 369ā“374

orthogonal polynomials, 542

uncountable, 17, 385, 445

parallelogram law, 594ā“596 uniqueness, 380ā“381

612 A COMPANION TO ANALYSIS

Riemann integral, see integral dominated convergence, 84

Riemann-Lebesgue lemma, 566 equivalent to sequences, 68, 287

Rolleā™s theorem Fubiniā™s theorem, 90

examination of proof, 453 monotone convergence, 470

interesting use, 63ā“64 rearranged, 81, 86, 467

statement and proof, 61ā“63 sup norm, 276

Routhā™s rule, 485 supremum

routine, 50 and fundamental axiom, 37

Russellā™s paradox, 375 deļ¬nition, 32

existence, 33

saddle, 157 use, 34ā“37

sandwich lemma, 7 surjective function, 475

Schur complement, 485 symmetric

Schwarz, area counterexample, 229 linear map, 481

Shannonā™s theorem, 236ā“241 matrix, diagonalisable, 446

Simpsonā™s rule, 496

singular points, see critical points Taylor series, see power series

slide rule, 111 Taylor theorems

solution of linear equations via best for examination, 189

Gauss-Siedel method, 590 Cauchyā™s counterexample, 143

Jacobi method, 590 depend on fundamental axiom, 145

global in R, 142, 189, 455

sovereigns, golden, 81ā“82

in R, 141ā“145

space ļ¬lling curve, 550

spaces little practical use, 190, 297

local in R, 142

C([a, b]), 1 ), 266, 278

local in Rn , 150ā“151, 154

C([a, b]), 2 ), 267, 278

C([a, b]), ā ), 278 Tchebychev

C([a, b]), p ), 533 inequality, 222

c0 , 270 polynomials, 454ā“456

l1 , 267 spelling, 455

l2 , 531 term by term

lā , 270 diļ¬erentiation, 287, 291, 302

lp , 533 integration, 287

s00 , 265 Thorā™s drinking horn, 522

L(U, U ), 587 Torricelliā™s trumpet, 521

L(U, V ), 253 total boundedness, 273

spectral radius, 588ā“590 total variation, 517

squeeze lemma, 7 transcendentals, existence of

Stirlingā™s formula, simple versions, 209, Cantorā™s proof, 385

238, 504 Liouvilleā™s proof, 435

successive approximation, 329ā“331 Trapezium rule, 495

successive bisection, see lion hunting trigonometric functions, 98ā“102, 143, 318,

summation methods, 461ā“464 519ā“520

sums, see also power series, Fourier se- troublesome operations, 306

ries, term by term and conver-

uniform

gence tests

continuity, 65ā“66, 182, 275

absolute convergence, 69

conditionally convergent, 78 convergence, 280ā“288

convergence, 68 norm, 275

613

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

uniqueness

antiderivative, 20, 186

completions, 356ā“358

decimal expansion, 13

Fourier series, 299

limit, 4, 47, 244

power series, 293

reals, 380ā“381

solution of diļ¬erential equations, 305ā“

308

universal chord theorem, 441

variation of parameters, 583

Vietaā™s formula for Ļ, 475

Vitaliā™s paradox, 171

volume of an n-dimensional sphere, 233

Wallis

formula for Ļ, 472

integrals of powers, 494

Weierstrass

M-test, 288

non-existence of minima, 199ā“202, 536ā“

538

polynomial approximation, 540

well ordering of integers, 10, 31

witchā™s hat

ordinary, 281

tall, 283

Wronskian, 320ā“321, 581ā“582

young man, deep, 326, 528

young woman, deep, 384

Zeno, 25ā“29

zeta function, brief appearance, 298

ńņš. 19 |