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A 1
¬B · ¬·
¬ · = M ’1 ¬0· ,
C   0
D 0

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

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

y (t) + 2βy (t) + ω 2 y(t) = f (t),

where β and ω are strictly positive real numbers and y : [0, ∞) ’ R satis¬es
y(0) = y (0) = 0. Use Green™s function methods to show that, if β > ω,
t
’1
f (s)e’β(t’s) sinh(±(t ’ s)) ds
y(t) = ±
0

where ± is the positive square root of β 2 ’ ω 2 and obtain a similar result in
the cases when ω > β and ω = β.
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 β which
causes y to die away as fast as possible. (Interpreting this last phrase in a
reasonable manner is part of your task.)
587
Please send corrections however trivial to twk@dpmms.cam.ac.uk

Exercise K.281. [13.1, T,] (Although this sequence of exercises seems to
¬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 ∈ 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 =
∞ ’1
and I ’ S ¤ T (1 ’ T )’1 .
j
j=0 T . Show also that S ¤ (1 ’ T )
(ii) We continue with the notation and assumptions of (i). By looking at
Sn (I ’ T ), show that S(I ’ T ) = I. Show also that (I ’ T )S = I. Conclude
that, if A ∈ L(U, U ) and I ’ A < 1, then A is invertible and
A’1 ¤ (1 ’ I ’ A )’1 and I ’ A’1 ¤ I ’ A (1 ’ I ’ A )’1 .
(iii) Suppose that B ∈ L(U, U ) is invertible and B’C < B ’1 ’1 . If we
set A = B ’1 C, show that A = B ’1 C is invertible. Show that A’1 B ’1 C = I
and CA’1 B ’1 = CBB ’1 A’1 B ’1 = I, and so C is invertible with inverse
A’1 B ’1 . Show, further that,
C ’1 ¤ B ’1 (1 ’ B ’1 B ’ C ) and
B ’1 ’ C ’1 ¤ B ’1 B ’ C (1 ’ B ’1 B ’ C ).
(iv) Let E be the set of invertible C ∈ L(U, U ). Show that E is open in
). Show further that, if we de¬ne ˜ : E ’ E by ˜(C) = C ’1 ,
(L(U, U ),
then ˜ is a continuous function.
(v) Returning to the discussion in (i) and (ii) show that if T < 1 then
(I ’ T )’1 ’ I ’ T ¤ T (1 ’ T )’1
2


and
(I + T )’1 ’ I + T ¤ T (1 ’ T )’1 .
2


Conclude that
˜(I + T ) = I ’ T + (T ) T ,
where : L(U, U ) ’ L(U, U ) is such that (T ) ’ 0 as T ’ 0. Con-
clude that ˜ is di¬erentiable at I. [If you wish to con¬ne yourself to ¬nite
dimensional spaces, take U ¬nite dimensional, but there is no need.]
(vi) Show that ˜ is everywhere di¬erentiable on E with D˜(B) = ¦B ,
where ¦B : L(U, U ) ’ L(U, U ) is the linear map given by
¦B (S) = ’B ’1 SB ’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, ‘ ] We continue with the
ideas and notation of Exercise K.281.
(i) Suppose that A ∈ L(U, U ). Show that

Ajk+r ¤ Aj k
A r.

= inf n≥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
’ .

We call the spectral radius of A and write ρ(A) = .
(iii) Show that ∞ An converges in the operator norm if ρ(A) < 1 and
n=0
diverges if ρ(A) > 1.
(iv) If ρ(A) < 1 show that I ’ A is invertible and (I ’ A)’1 = ∞ An .
n=0

Exercise K.283. [13.1, T] We continue with the ideas of Exercise K.282.
(i) If ± ∈ L(U, U ) and » is a scalar show that ρ(»±) = |»|ρ(±).
(ii) Write down a linear map β : Cm ’ Cm such that β m’1 = 0 but
β m = 0. What is the value of ρ(β)?
(iii) If the linear map ± : Cm ’ Cm has m distinct eigenvalues explain
brie¬‚y why we can ¬nd an invertible linear map θ : Cm ’ Cm such that θ±θ ’1
has a diagonal matrix D with respect to the standard basis. By considering
the behaviour of D n , or otherwise, show that

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

(For an improvement of this result see Exercise K.285.)
(iv) Give two linear maps ±, β : C2 ’ C2 such that ρ(±) = ρ(β) = 0 but
ρ(± + β) = 1.
Exercise K.284. (Cayley-Hamilton.) This question requires you to know
that any linear map ± : Cm ’ Cm has an upper triangular matrix with re-
spect to some basis. Recall also that, if ± : Cm ’ Cm we can de¬ne its
characteristic polynomial χ± by the condition det(tι ’ ±) = χ± (t) for all
t ∈ C. (Here and elsewhere, ι : Cm ’ Cm is the identity map.)
(i) We work with the standard basis on Cm . Explain why, if ± : Cm ’ Cm
is linear, we can ¬nd an invertible linear map θ : Cm ’ Cm such that
β = θ±θ’1 has an upper triangular matrix with respect to the standard
basis.
(ii) Show that, if β : Cm ’ Cm has an upper triangular matrix with
respect to the standard basis and > 0, then we can ¬nd γ : Cm ’ Cm with
m distinct eigenvalues such that γ ’ β < .
589
Please send corrections however trivial to twk@dpmms.cam.ac.uk

(iii) Deduce that, if ± : Cm ’ Cm is linear we can ¬nd linear ±k : Cm ’
Cm with m distinct eigenvalues and

±k ’ ± ’ 0 as k ’ ∞.

(iv) (Pure algebra) If ± : Cm ’ Cm has n distinct eigenvalues, show, by
considering the matrix of ± with respect to a basis of eigenvectors that

χ± (±) = 0.

(v) Show carefully, using (iii) and limiting arguments, that, if ± : C m ’
Cm is linear, then

χ± (±) = 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, ‘] We continue with the ideas of Exercise K.283.
(i) If ±, β ∈ L(U, U ) and ± and β commute, show that

ρ(± + β) ¤ ρ(±) + ρ(β).

(Compare Exercise K.283 (iv).)
(ii) By using the ideas of Exercise K.284, or otherwise, show that, if
± : Cm ’ Cm is linear and > 0, we can ¬nd a β : Cm ’ Cm such that
β < , ± and β commute and ± + β has m distinct eigenvalues.
(iii) By using (ii), or otherwise, show that, if ± : Cm ’ Cm is linear, then

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

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

Exercise K.286. [13.1, T, ‘] In previous questions we have shown that, if
± : Cm ’ Cm is linear, and we write

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

then ±n 1/n ’ ρ(±).
Suppose that b ∈ Cm , we choose an arbitrary x0 ∈ Cm and we consider
the sequence

xn+1 = b + ±xn .
590 A COMPANION TO ANALYSIS

If ρ(±) < 1, give two proofs that
xn ’ (ι ’ ±)’1 b ’ 0 as n ’ ∞.
(i) By ¬nding xn explicitly and using Exercise K.282.
(ii) By using the ¬rst paragraph of Exercise K.261.
By considering an eigenvector corresponding to an eigenvalue of largest
modulus show that the sequence will diverge if ρ(±) > 1 and we choose x0
suitably. Show that, if ρ(±) = 1, either we can ¬nd an x0 such the sequence
diverges or, if the sequence always converges, we can ¬nd two starting points
with di¬erent limits.
Exercise K.287. [13.1, T, ‘ ] Consider the problem of solving the equation
Ax = b
where A is an m — 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 coe¬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 coe¬cients in I ’ A in a well organized pattern.
(Such problems arise in the numerical solution of important partial di¬eren-
tial equations.) The following method can then be employed. Choose x0 and
de¬ne a sequence
xn+1 = b + (I ’ 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 e¬cient.
The rest of the question consists of elaboration of this idea. We require
Exercise K.286. Suppose that A is an m — m matrix and A = D ’ U ’ 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 = D’1 (b + (U + L)xn ),
Gauss-Seidel xn+1 = (D ’ L)’1 (b + U xn ).
591
Please send corrections however trivial to twk@dpmms.cam.ac.uk

For each of these two schemes give a necessary and su¬cient condition in
terms of the spectral radius of an appropriate matrix for the method to
work.
Another iterative scheme uses

xn+1 = (D ’ ωL)’1 (ωb + ((1 ’ ω)D + ωU )xn ),

where ω is some ¬xed real number. Give a necessary and su¬cient condition,
in the form ρ(H) < 1 where H is an appropriate matrix, for the method to
work. By showing that

det H = (1 ’ ω)m ,

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

Exercise K.288. [13.1, S, ‘‘ ] (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. ˜Moreover Df |’1 is continuous on
B
B.™

Exercise K.289. [13.1, T, ‘‘ ] This is another exercise in the ideas of Ex-
ercise K.281. We work in (U, U ) as before.
(i) Show that, if ± ∈ L(U, U ), we can ¬nd exp ± ∈ L(U, U ) such that

n
±r
’ exp ± ’ 0
r!
r=0

as n ’ ∞.
(ii) Show carefully that, if ± and β commute,

exp ± exp β = exp(± + β).

(iii) Show that if ± and β are general (not necessarily commuting) ele-
ments of L(U, U ), then

h’2 (exp(h±) exp(hβ) ’ exp(hβ) exp(h±)) ’ (±β ’ β±) ’ 0

as the real number h ’ 0.
Conclude that, in general, exp(±) exp(β) and exp(β) exp(±) need not be
equal. Deduce also that exp(± + β) and exp(±) exp(β) need not be equal.
(iv) Show carefully (you must bear part (iii) in mind) that exp : L(U, U ) ’
L(U, U ) is everywhere continuous.
592 A COMPANION TO ANALYSIS

Exercise K.290. [13.1, P, ‘ ] (i) Consider the map ˜3 : L(U, U ) ’ L(U, U )
given by ˜3 (±) = ±3 . Show that ˜ is everywhere di¬erentiable with
D˜3 (±)β = β±2 + ±β±2 + ±2 β.
(ii) State and prove the appropriate generalisation to the map ± ’ ± m
with m a positive integer.
(iii) Show that exp, de¬ned in Exercise K.289, is everywhere di¬erentiable.
[This requires care.]
Exercise K.291. [13.1, P] Suppose U is a ¬nite dimensional vector space
over C. Let ± : U ’ U be a linear map. If ± has matrix representation A
with respect to some basis, explain why, as N ’ ∞, the entries of the matrix
N n
n=0 A /n! converge to the entries of a matrix exp A which represents exp ±
with respect to the given basis. (See Exercise K.289.)
It is a theorem that any ± ∈ L(U, U ) has an upper triangular matrix with
respect to some basis. By using this fact, or otherwise, show that
det(exp ±) = eTrace ± .
Exercise K.292. [13.1, P] We work in the space M2 (R) of 2 — 2 real ma-
trices. We give M2 (R) the associated operator norm.
(i) Show that the map S : M2 (R) ’ M2 (R) given by S(A) = A2 is
everywhere di¬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 = ’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
593
Please send corrections however trivial to twk@dpmms.cam.ac.uk

has exactly one solution of the form A = I + Y with Y ∈ V for each X ∈ U .
(v) Show that we can not ¬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 ’1

with Y ∈ V for each X ∈ 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 ¬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 ∈ V for each X ∈ 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 ± : R3 ’ R3 is an antisymmetric linear map (that is
±T = ’±). Show that x and ±x are orthogonal and that the eigenvalues of
±2 must be non-positive real numbers. By considering eigenvectors of ± and
±2 show that we can always ¬nd µ ∈ R and three orthonormal vectors e1 , e2
and e3 , such that

±e1 = µe2 , ±e2 = ’µe1 , ±e3 = 0.

By choosing appropriate axes and using matrix representations show that
exp ± is a rotation.

Exercise K.294. [13.1, P, G] Show that every linear map ± : Rm ’ Rm
is the sum of a symmetric and an antisymmetric linear map. Suppose ± is
an orthogonal map (that is, ±±T = ι) with ι ’ ± < with very small.
Show that
2
±=ι+ β+ γ

with β , γ ¤ 2.
594 A COMPANION TO ANALYSIS

Exercise K.295. [13.3, M] (Treat this as a ˜methods question™.) 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 2θ1 = ∠AOB, 2θ2 = ∠BOC, 2θ3 = ∠COD,
2θ4 = ∠DOA. Find the area of the quadrilateral and state the relation that
θ1 , θ2 , θ3 and θ4 must satisfy.
Use Lagrange™s method to ¬nd the form of a quadrilateral of greatest area
inscribed in a circle of radius a. (Treat this as a ˜methods question™.)
Use Lagrange™s method to ¬nd the form of an n-gon of greatest area
inscribed in a circle [n ≥ 3].
Use Lagrange™s method to ¬nd the form of an n-gon of least area circum-
scribing a circle [n ≥ 3].
[Compare Exercise K.40.]

Exercise K.296. [13.3, T] Let p and q be strictly positive real numbers
with p’1 + q ’1 = 1. Suppose that y1 , y2 , . . . , yn , c > 0. Explain why there
must exist x1 , x2 , . . . , xn ≥ 0 with n xp = c and
j=1 j

n n n
tp = c.
xj yj ≥ tj yj whenever t1 , t2 , . . . , tn ≥ 0 with j
j=1 j=1 j=1

Use the Lagrange multiplier method to ¬nd the xj . Deduce from your
answer that
1/p 1/q
n n n
|aj |p |bj |q
|aj bj | ¤
j=1 j=1 j=1

whenever aj , bj ∈ C. Under what conditions does equality hold?
This gives an alternative proof of the ¬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 ∈ V . Show that
2 2 2 2
+ x’y
x+y =2 x +2 y

for all x, y ∈ V . (This is called the parallelogram law.)
(ii) Show also that
2 2
’ x’y
4 x, y = x + y

for all x, y ∈ V . (This is called the polarisation identity.)
595
Please send corrections however trivial to twk@dpmms.cam.ac.uk

(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 ’ y | ¤ x ’ 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
’ i x ’ iy 2 .
’ x’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 ∈ C([0, 1]).


Exercise K.298. [14.1, T, ‘ ] 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
+ x’y
x+y =2 x +2 y

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

x, y = 4’1 2 2
’ x’y
x+y .

Show that x, y = y, x for all x, y ∈ V and that x, x = x 2 (so that,
automatically, x, x ≥ 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+v’w
u+v+w =2 u+v +2 w

and use this to establish that

u + w, v + u ’ w, v = 2 u, v (1)

for all u, v, w ∈ 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 ∈ V .
(iii) Establish the equation
»x, y = » x, y
for all positive integer values of », then for all integer values, for all rational
values and then for all real values of ».
(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 ’E y E for all x, y ∈
E. Suppose further that there is map ME : E 2 ’ 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) (»x)y = »(xy), x(»y) = »(xy),
(iv) xy ¤ x y ,
for all x, y, z ∈ E and » ∈ F. Show that, if X is given the structure of a
normed vector space as in Lemma 14.1.9, then we can ¬nd a map M : X 2 ’
X such that M (x, y) = ME (x, y) for all x, y ∈ 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 ∈ E, then M (x, y) = M (y, x) for all x, y ∈ X. Show that, if there
exists an e ∈ E such that ME (x, e) = ME (e, x) = x for all x ∈ E, then
M (x, e) = M (e, x) = x for all x ∈ 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 ∈ E. For each e ∈ E de¬ne fe (t) = d(e, t) ’ d(e0 , t) [t ∈ E].
Show that fe ∈ C(E), where C(E) is the space of bounded continuous function
g : E ’ R.
(ii) Give C(E) the usual uniform norm. Show that
fu ’ fv = d(u, v)
for all u, v ∈ E.
(iii) Let Y be the closure of {fe : e ∈ E}. By using Theorem 11.3.7, or
˜
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 θ : E ’ Y given
by θ(e) = fe .
14
I take it from from a book [15] crammed with neat proofs.
597
Please send corrections however trivial to twk@dpmms.cam.ac.uk

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 ’ R given by

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

de¬nes a metric on X 2 . Show that, if (X, d) is complete, so is (X 2 , d2 ).
(ii) Consider X = [0, ∞) with the usual Euclidean metric d and E =
(0, ∞). 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 ) ’ (E, d) is continuous. Show, however,
that there does not exist a continuous map M : (X 2 , d2 ) ’ (X, d) with
M (x, y) = ME (x, y) for all x, y ∈ E such that (X, M ) is a group.
(iii) Consider X = [0, ∞) with the usual Euclidean metric d and E =
(0, ∞) © 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 ) ’ (E, d)
is continuous. Show, however, that there does not exist a continuous map
M : (X 2 , d2 ) ’ (X, d) with M (x, y) = ME (x, y) for all x, y ∈ E such that
(X, M ) is a group.
(iv) Consider X = (0, ∞). Show that if we write

d(x, y) = | log x ’ log y|

then (X, d) is a metric space. Let E = (0, ∞) © 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 ) ’ (E, d) is continuous. Show that there
exists a continuous map M : (X 2 , d2 ) ’ (X, d) with M (x, y) = ME (x, y) for
all x, y ∈ 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 = {n± : n ∈ Z}

for some ± ∈ R.
(ii) Observe that

S 1 = {» ∈ C : |»| = 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, ‘‘ ] Exercise K.56 is not important in itself, but
the method of its proof is. Extend the result and proof to f : E ’ R where
E is a dense subspace of a metric space (X, d).
Can the result be extended to f : E ’ Y where E is a dense subspace
of a metric space (X, d) and (Y, ρ) is a metric space? (Give a proof or
counterexample.) If not, what natural extra condition can we place on (Y, ρ)
so that the result can be extended?

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

|f (x) ’ f (y)| ¤ (x ’ y)2 for all x, y ∈ 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 ¬nd a ¬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 . De¬ne ρ : R2 — R2 ’ R by

1 if y1 = y2
ρ((x1 , y1 ), (x2 , y2 )) =
min(1, |x1 ’ x2 |) if y1 = y2 .

Show that ρ is a metric and that ρ is complete. Show that, if we work in
(R2 , ρ),

V = {(x, y) : |x| < 1, y ∈ 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 ¬nd bounded closed sets Kj with
Kj+1 ⊆ Kj [1 ¤ j] and ∞ 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 ∈ U , write x ∼ y if there is an
open interval (a, b) ‚ U with x, y ∈ (a, b). Show that ∼ is an equivalence
relation on U .
(ii) Write [x] = {y ∈ U : y ∼ x} for the equivalence class of some x ∈ U .
If [x] is bounded, show, by considering the in¬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, ∞), (’∞, c) and R.
(iv) Suppose that U is the disjoint union of a collection C of sets of the
form (a, b), (c, ∞), (’∞, c) and R. If J ∈ C explain why there exists an
I ∈ C with I ⊇ J. Explain why, if a is an end point of J which is not an end
point of I, there must exist a J ∈ C with a ∈ 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 de¬ned and this raises the
possibility of de¬ning the ˜length™ 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 (’a, a) — (’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, ρ) are
homeomorphic if there exists a bijective map f : X ’ Y such that f and
f ’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 ’ Y is a homeomorphism, show that U is open in (X, d) if
and only if f (U ) is open in (Y, ρ).
(iii) By constructing an explicit homeomorphism, show that R with the
usual metric is homeomorphic to the open interval (’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 = (’1, 1)
with the usual metric is homeomorphic to

J = {z ∈ C : |z| = 1, z = 1}

with the usual metric. Show that [’1, 1] with the usual metric is a completion
of I. Find a completion of J.
Explain brie¬‚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, ρ) is a any metric space. If f : X ’ Y
is a bijective continuous function show that (Y, ρ) has the Bolzano“Weierstrass
property and that f ’1 : Y ’ X is uniformly continuous. (Note that we have
shown that, in the language of Exercise K.308, (X, d) and (Y, ρ) are homeo-
morphic.)
(ii) Look brie¬‚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 ’ R with f ’1 not uniformly continuous.
(iv) Let d be the usual metric on R and ρ the discrete metric on R. Let
f : (R, ρ) ’ (R, d) be given by f (x) = x. Show that f is is a bijective
continuous function but f ’1 is not continuous.
601
Please send corrections however trivial to twk@dpmms.cam.ac.uk

Exercise K.310. [14.1, T, ‘ ] 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 ¬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 ’ l2 given by

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

is well de¬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, ρ) are metric spaces
and f : X ’ Y is a continuous surjective map.
(i) Suppose that ρ(f (x), f (x )) ¤ Kd(x, x ) for all x, x ∈ X and some
K > 0. If (X, d) is complete, does it follow that (Y, ρ) is complete? If
(Y, ρ) is complete, does it follow that (X, d) is complete? Give proofs or
counterexamples as appropriate.
(ii) Suppose that ρ(f (x), f (x )) ≥ Kd(x, x ) for all x, x ∈ X and some
K > 0. If (X, d) is complete, does it follow that (Y, ρ) is complete? If
(Y, ρ) 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 ± = (a, b)
with a < b in R. If ±, β ∈ X, then the symmetric di¬erence ± β consists
of the empty set, one open interval or two disjoint open intervals. We de¬ne
d(±, β) to be the total length of the intervals making up ± β. 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 ∈ N and n = 0, then there exist unique r, s ∈ N with s an odd integer and
n = s2r . Write »(n) = r. If n, m ∈ N and n = m we write

d(n, m) = 2’»(|n’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 ’ N given by f (x) = x2 is continuous.
(v) Show that the function g : N ’ 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.)
Re¬‚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 ’ R which is in¬nitely di¬erentiable at at every point except
0, which has directional derivative zero in all directions at 0 but which is not
di¬erentiable at 0.
Bibliography

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[2] A. F. Beardon. Limits. A New Approach to Real Analysis. Springer,
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[3] R. Beigel. Irrationality without number theory. American Mathematical
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[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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[12] W. S. Churchill. My Early Life. Thornton Butterworth, London, 1930.

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[14] J. Dieudonn´. In¬nitesimal Calculus. Kershaw Publishing Company,
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604 A COMPANION TO ANALYSIS

[15] R. M. Dudley. Real Analysis and Probability. Wadsworth and Brooks,
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derivatives. American Mathematical Monthly, 105:756“8, 1998.

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a

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[45] K. R. Stromberg. Introduction to Classical Real Analysis. Wadsworth,
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1902. Still available in its 4th edition.

[48] P. Whittle. Optimization under Constraints. Wiley, 1971.
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, 169“172, 214“217, binomial theorem, 297, 562
229“231 bisection, bisection search, see lion hunt-
authors, other ing
Beardon, 56, 395 Bishop™s constructive analysis, 415“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, 38“39
Burn, 62
for closed bounded sets in Rm , 49
Conway, 449
Dieudonn´, viii, 25, 60, 154, 206
e for metric spaces, 272“274
in Rm , 47
Halmos, 375, 391
Hardy, viii, 43, 83, 103, 154, 297 bounded variation, functions of, 181, 516“
Klein, 113, 422 519
Kline, 375 brachistochrone problem, 190
Littlewood, 81
Petard, H., 16 calculus of variations
Plato, 28 problems, 198“202
Poincar´, 376
e successes, 190“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 ¬eld, 379 function not given by Taylor series,
143
general discussion, 242, 364“365, 375“
377 mean value theorem, 457
Zermelo-Fraenkel, 375 proof of binomial theorem, 562
sequence, 67, 263
balls solution of di¬erential equations, 563
open and closed, 50, 245 Cauchy-Riemann equations, 479
packing, 233“235 Cauchy-Schwarz inequality, 44
packing in Fn , 237“238 Cayley-Hamilton theorem, 588
2
Banach, 242, 303 chain rule
Bernstein polynomial, 542 many dimensional, 131“132


607
608 A COMPANION TO ANALYSIS

one dimensional, 102“104 convergence tests for sums
Chebychev, see Tchebychev Abel™s, 79
chords, 475 alternating series, 78
closed bounded sets in Rm Cauchy condensation, 76
and Bolzano-Weierstrass, 49 comparison, 70
and continuous functions, 56“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
de¬nition for Rm , 49 function, 451, 498“499
de¬nition for metric space, 244 set, 447, 571
key properties, 52, 245 convolution, 565“566
closure of a set, 526 countability, 383“386
comma notation, 126, 148 critical points, see also maxima and min-
compactness, 421, 536 ima, 154“160, 163“167, 340
completeness
D notation, 126, 150
de¬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, 355“358
derivative
existence, 362“364, 596
complex, 288“289
ordered ¬elds, 411“413
directional, 125
structure carries over, 358“361
general discussion, 121“127
unique, 356“358
in applied mathematics, 152, 401“
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, 367“368 more general, 253
Q from Z, 366“367 not continuous, 452
R from Q, 369“374 partial, 126
Z from N, 366 devil™s staircase, 552
diagrams, use of, 98
continued fractions, 436“440
di¬erential equations
continuity, see also uniform continuity
and Green™s functions, 318“326
discussed, 7, 388“391, 417
and power series, 294, 563
of linear maps, 128, 250“253
Euler™s method, 577“580
pointwise, 7, 53, 245
existence and uniqueness of solutions,
via open sets, 54, 246
305“318
continuous functions
di¬erentiation
exotic, 2, 549“554
Fourier series, 302
integration of, 182“186
on closed bounded sets in Rm , 56“59 power series, 291, 557“558
term by term, 291
continuum, models for, see also reals and
under the integral
rationals, 25“28, 418“419
¬nite range, 191
contraction mapping, 303“305, 307, 330,
408 in¬nite range, 287
609
Please send corrections however trivial to twk@dpmms.cam.ac.uk

Dini™s theorem, 542 Greek rigour, 29, 365, 376, 521
directed set, 396 Green™s functions, 318“326, 583“586
dissection, 172
Hahn-Banach for Rn , 447
dominated convergence
Hausdor¬ metric, 534
for some integrals, 547
for sums, 84 Heine-Borel theorem, 449
duck, tests for, 369 Hessian, 157
hill and dale theorem, 164“166
economics, fundamental problem of, 58 H¨lder™s inequality, 531, 533
o
escape to in¬nity, 84, 283“284 homeomorphism, 600
Euclidean homogeneous function, 480
geometry, 364“365
norm, 44 implicit function theorem
Euler discussion, 339“347
method for di¬erential equations, 577“ statement and proof, 343“344
580 indices, see powers
on homogeneous functions, 480 inequality
Euler™s γ, 467 arithmetic-geometric, 451
Euler-Lagrange equation, 194 Cauchy-Schwarz, 44
exponential function, 91“98, 143, 317, 417, H¨lder™s, 531, 533
o
497, 591 Jensen™s, 450, 498
extreme points, 447“448 Ptolomey™s, 443
reverse H¨lder, 532
o
Father Christmas, 172
Tchebychev, 222
¬xed point theorems, 17, 303“304
in¬mum, 34
Fourier series, 298“302
in¬nite
Fubini™s theorem
products, 472, 561
for in¬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, 477“479
for l2 , 531
fundamental axiom, 9
for Rn , 43
fundamental theorem of algebra
integral kernel, example of, 326
proof, 114“117
integral mean value theorem, 490
statement, 113
integrals
theorem of analysis, 114, 120
along curves, 228“229, 231
fundamental theorem of the calculus
and uniform convergence, 282
discussion of extensions, 186
improper (or in¬nite), 207“211
in one dimension, 184“186
of continuous functions, 182“186
over area, 212“217
Gabriel™s horn, 521
principle value, 211
Gaussian quadrature, 544
Riemann, de¬nition, 172“174
general principle
Riemann, problems, 205“206, 214
of convergence, 68, 263, 412
Riemann, properties, 174“181
of uniform convergence, 280
Riemann-Stieltjes, 217“224, 519
generic, 164
Riemann-Stieltjes, problems, 220
geodesics, 254“260
vector-valued, 202“204
global and local, contrasted, 65, 123, 142“
144, 155, 160, 164, 314“317, 341 integration
610 A COMPANION TO ANALYSIS

by parts, 189 method, 350“351
by substitution, 187 necessity, 350
numerical, 495“497, 544 su¬ciency, 353
Riemann versus Lebesgue, 206“207 Leader, examples, 528
term by term, 287 left and right derivative, 424
interchange of limits Legendre polynomials, 544“545, 559
derivative and in¬nite integral, 287 Leibniz rule, 580
derivative and integral, 191 limits
general discussion, 83“84 general view of, 395“400
in¬nite integrals, 512 in metric spaces, 243
integral and sum, 287 in normed spaces, 244
integrals, 213 more general than sequences, 55“56
limit and derivative, 285, 286 pointwise, 280
sequences in Rm , 46
limit and integral, 282“284
limit and sum, 84 sequences in ordered ¬elds, 3“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, 15“16, 58, 491“492
false for rationals, 2 in Rm , 48
not available in constructive analy- Lipschitz
sis, 418 condition, 307
obvious?, 25“28 equivalence, 248
true for reals, 15 logarithm
international date line, 108 for (0, ∞), 104“106, 476, 497
inverse function theorem non-existence for C \ {0}, 108“109,
alternative approach, 407“410 315“317
gives implicit function theorem, 342 what preceded, 475
many dimensional, 337
one dimensional, 106, 402 Markov chains, 571“574
inverses in L(U, U ), 336, 587 maxima and minima, 58, 154“160, 194“
irrationality of 202, 347“354
e, 97 Maxwell
√ 467
γ?, hill and dale theorem, 164
2, 432 prefers coordinate free methods, 46,
irrelevant m, 269 121
isolated points, 263 mean value inequality
for complex di¬erentiation, 289
Jacobian
for reals, 18“20, 22, 36, 60
determinant, 406
many dimensional, 136“138
matrix, 127
mean value theorem
Jensen™s inequality, 450, 498
Cauchy™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, 278“279 partial derivatives
British railway non-stop, 243 and Jacobian matrix, 127
British railway stopping, 243 and possible di¬erentiability, 147, 161
complete, 263 de¬nition, 126
completion, 363 notation, 126, 148, 150, 401“404, 423
de¬nition, 242 symmetry of second, 149, 162
derived from norm, 242 partition, see dissection
discrete, 273 pass the parcel, 339
Hausdor¬, 534 piecewise de¬nitions, 425
Lipschitz equivalent, 248 placeholder, 241, 350, 422
totally bounded, 273 pointwise compared with uniform, 65, 280,
M¨bius transformation, 255“261
o 282
monotone convergence power series
for sums, 470 addition, 459
and di¬erential equations, 294, 563
neighbourhood, 50, 245 composition, 470
non-Euclidean geometry, 364“365 convergence, 71
norm di¬erentiation, 291, 557“558
all equivalent on Rn , 248 limitations, 143, 298
completion, 358, 364 many variable, 469
de¬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
de¬nition of, 109“113, 294“296, 555“
and · , 241
557
ι, 588
primary schools, Hungarian, 384
x, y , 43
primes, in¬nitely many, Euler™s proof, 473
g,ij , 148
probability theory, 221“224, 240“241
g,j , 126
Ptolomey™s inequality, 443
z — , 119
x · y, 43
quantum mechanics, 27
non-uniform, 422“423
nowhere di¬erentiable continuous func-
radical reconstructions of analysis, 375,
tion, 549
415“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
de¬nition for Rm , 50
not good for analysis, 1“3
de¬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, 369“374
orthogonal polynomials, 542
uncountable, 17, 385, 445
parallelogram law, 594“596 uniqueness, 380“381
612 A COMPANION TO ANALYSIS

Riemann integral, see integral dominated convergence, 84
Riemann-Lebesgue lemma, 566 equivalent to sequences, 68, 287
Rolle™s theorem Fubini™s theorem, 90
examination of proof, 453 monotone convergence, 470
interesting use, 63“64 rearranged, 81, 86, 467
statement and proof, 61“63 sup norm, 276
Routh™s rule, 485 supremum
routine, 50 and fundamental axiom, 37
Russell™s paradox, 375 de¬nition, 32
existence, 33
saddle, 157 use, 34“37
sandwich lemma, 7 surjective function, 475
Schur complement, 485 symmetric
Schwarz, area counterexample, 229 linear map, 481
Shannon™s theorem, 236“241 matrix, diagonalisable, 446
Simpson™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 Cauchy™s counterexample, 143
Jacobi method, 590 depend on fundamental axiom, 145
global in R, 142, 189, 455
sovereigns, golden, 81“82
in R, 141“145
space ¬lling curve, 550
spaces little practical use, 190, 297
local in R, 142
C([a, b]), 1 ), 266, 278
local in Rn , 150“151, 154
C([a, b]), 2 ), 267, 278
C([a, b]), ∞ ), 278 Tchebychev
C([a, b]), p ), 533 inequality, 222
c0 , 270 polynomials, 454“456
l1 , 267 spelling, 455
l2 , 531 term by term
l∞ , 270 di¬erentiation, 287, 291, 302
lp , 533 integration, 287
s00 , 265 Thor™s drinking horn, 522
L(U, U ), 587 Torricelli™s trumpet, 521
L(U, V ), 253 total boundedness, 273
spectral radius, 588“590 total variation, 517
squeeze lemma, 7 transcendentals, existence of
Stirling™s formula, simple versions, 209, Cantor™s proof, 385
238, 504 Liouville™s proof, 435
successive approximation, 329“331 Trapezium rule, 495
successive bisection, see lion hunting trigonometric functions, 98“102, 143, 318,
summation methods, 461“464 519“520
sums, see also power series, Fourier se- troublesome operations, 306
ries, term by term and conver-
uniform
gence tests
continuity, 65“66, 182, 275
absolute convergence, 69
conditionally convergent, 78 convergence, 280“288
convergence, 68 norm, 275
613
Please send corrections however trivial to twk@dpmms.cam.ac.uk

uniqueness
antiderivative, 20, 186
completions, 356“358
decimal expansion, 13
Fourier series, 299
limit, 4, 47, 244
power series, 293
reals, 380“381
solution of di¬erential equations, 305“
308
universal chord theorem, 441

variation of parameters, 583
Vieta™s formula for π, 475
Vitali™s paradox, 171
volume of an n-dimensional sphere, 233

Wallis
formula for π, 472
integrals of powers, 494
Weierstrass
M-test, 288
non-existence of minima, 199“202, 536“
538
polynomial approximation, 540
well ordering of integers, 10, 31
witch™s hat
ordinary, 281
tall, 283
Wronskian, 320“321, 581“582

young man, deep, 326, 528
young woman, deep, 384

Zeno, 25“29
zeta function, brief appearance, 298

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