Exercise K.187. [11.1, P] Consider the space s00 introduced in Exer-

cise 10.4.8. Recall that s00 the space of real sequences a = (an )∞ such

n=1

that all but ¬nitely many of the an are zero. Our object in this question is

to show that no norm on s00 can be complete. To this end, let us suppose

that is a norm on s00 .

(i) Write En = {a ∈ s00 : aj = 0 for all j ≥ n}. Show that En is closed.

(ii) If b ∈ En+1 \ En , show that there exists a δ > 0 such that b ’ a > δ

for all a ∈ En .

(iii) If h ∈ En+1 \ En and a ∈ En , show that a + »h ∈ En+1 \ En for all

» = 0.

(iv) Show that, given x(n) ∈ En and δn > 0, we can ¬nd x(n + 1) ∈ En+1

such that x(n+1)’x(n) < δn /4, and δn+1 < δn /4 such that x(n+1)’a >

δn+1 for all a ∈ En .

(v) Starting with x(1) = 0 and δ1 = 1, construct a sequence obeying the

conclusions of part (iv) for all n ≥ 2. Show that the sequence x(n) is Cauchy.

(vi) Continuing with the notation of (v), show that, if the sequence x(n)

converges to some y ∈ s00 then y ’ x(n + 1) < δn+1 /2, for each n ≥ 1.

Conclude that y ∈ En for any n and deduce the required result by reductio

/

ad absurdum.

531

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

Exercise K.188. (The space l 2 .) [11.1, T] (i) Let aj , bj ∈ R. Use the

triangle inequality for the Euclidean norm on Rm and careful handling of

limits to show that, if ∞ a2 and ∞ b2 converge, so does ∞ (aj + bj )2 .

j=1 j j=1 j j=1

Show further that, in this case,

1/2 1/2 1/2

∞ ∞ ∞

(aj + bj )2 a2 b2

¤ + .

j j

j=1 j=1 j=1

∞

(ii) Show that the set l 2 of real sequences a with 2

j=1 aj convergent

forms a vector space if we use the natural de¬nitions of addition and scalar

multiplication

(an ) + (bn ) = (an + bn ), »(an ) = (»an ).

Show that, if we set

∞

a2 ,

a =

2 j

j=1

then (l2 , 2 ) is a complete normed space.

(iii) The particular space (l 2 , 2 ) has a further remarkable property to

which we devote the next paragraph.

Show using the Cauchy“Schwarz inequality for Rm , or otherwise, that if

a, b ∈ l2 , then ∞ |aj bj | converges. We may thus de¬ne

j=1

∞

a·b= aj b j .

j=1

Show that this inner product satis¬es all the conclusions of Lemma 4.1.1.

Exercise K.189. (H¨lder™s inequality.) [11.1, T] Throughout this ques-

o

tion p and q are real numbers with p > 1 and p’1 + q ’1 = 1.

(i) Show that q > 1.

(ii) Use the convexity of ’ log (see Question K.39) to show that, if x and

y are strictly positive real numbers, then

xp y q

xy ¤ +.

p q

Observe that this equality remains true if we merely assume that x, y ≥ 0.

(iii) Suppose that f, g : [a, b] ’ R are continuous. Show that

b b b

1 1

p

|g(t)|q dt.

|f (t)g(t)| dt ¤ |f (t)| dt +

p q

a a a

532 A COMPANION TO ANALYSIS

Deduce that, if F, G : [a, b] ’ R are continuous and

b b

p

|G(t)|q dt = 1,

|F (t)| dt =

a a

then

b

|F (t)G(t)| dt ¤ 1.

a

(iv) By considering κF and µG with F and G as in (iii), or otherwise,

show that, if f, g : [a, b] ’ R are continuous, then

1/p 1/q

b b b

|f (t)|p dt |g(t)|q dt

|f (t)g(t)| dt ¤ .

a a a

This inequality is known as H¨lder™s inequality for integrals.

o

(v) Show that the Cauchy-Schwarz inequality for integrals is a special

case of H¨lder™s inequality.

o

Exercise K.190. [11.1, T, ‘ ] (i) Suppose that f : [a, b] ’ R is a continuous

function such that

1/q

b b

|g(t)|q dt

|f (t)g(t)| dt ¤ A

a a

for some constant A and all continuous functions g : [a, b] ’ R. By taking

g(t) = f (t)± for a suitable ±, or otherwise, show that

1/p

b

|f (t)|p dt ¤ A.

a

This result is known as the ˜reverse H¨lder inequality™.

o

(ii) By ¬rst applying H¨lder™s inequality to the right hand side of the

o

inequality

b b b

|(f (t) + h(t))g(t)| dt ¤ |f (t)g(t)| dt + |h(t)g(t)| dt

a a a

and then using the reverse H¨lder inequality, show that

o

1/p 1/p 1/p

b b b

p p p

|f (t) + h(t)| dt ¤ |f (t)| dt |h(t)| dt

+

a a a

533

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

for all continuous functions f, h : [a, b] ’ R. (This inequality is due to

Minkowski.)

(iii) Show that, if we set

1/p

b

|f (t)|p dt

f = ,

p

a

then (C([a, b]), p ) is a normed space. Show, however, that (C([a, b]), p)

is not complete.

(iv) Rewrite H¨lder™s inequality and the reverse H¨lder inequality using

o o

the notation introduced in (iii).

Exercise K.191. The l p spaces.) [11.1, T, ‘ ] Throughout this question

p and q are real numbers with p > 1 and p’1 + q ’1 = 1.

(i) By imitating the methods of Exercise K.189, or otherwise, show that,

if aj , bj ∈ R, then

1/p 1/q

n n n

|aj |p |bj |q

|aj ||bj | ¤ .

j=1 j=1 j=1

∞ ∞

|aj |p and |bj |q converge, then so does

Deduce carefully that, if j=1 j=1

∞

j=1 |aj ||bj | and

1/p 1/q

∞ ∞ ∞

|aj |p |bj |q

|aj ||bj | ¤ .

j=1 j=1 j=1

This inequality is known as H¨lder™s inequality for sums.

o

(ii) By imitating the methods of Exercise K.189, or otherwise, show that

the collection lp of real sequences a = (a1 , a2 , . . . ) forms a normed vector

space under the appropriate algebraic operations (to be speci¬ed) and norm

1/p

∞

|aj |p

a = .

p

j=1

(iii) By using the ideas of Example 11.1.10, or otherwise, show that

p

(l , p ) is complete.

(iv) Use the parallelogram law (see Exercise K.297 (i)) to show that the

p

l norm is not derived from an inner product if p = 2. Do the same for

(C([a, b]), p ).

∞

(v) We de¬ned (l1 , 1 ) in Exercise 11.1.10 and (l , ∞ ) in Exer-

cise 11.1.13. Investigate H¨lder™s inequality and the reverse H¨lder inequality

o o

for p = 1, q = ∞ and for p = ∞, q = 0. Carry out a similar investigation for

(C([a, b]), 1 ) (see Lemma 11.1.8) and (C([a, b]), ∞ ).

534 A COMPANION TO ANALYSIS

Exercise K.192. (The Hausdor¬ metric.) [11.1, T] Let (X, d) be a

metric space and E a non-empty subset of X. For each x ∈ X, de¬ne

d(x, E) = inf{d(x, y) : y ∈ E}.

Show that the map f : X ’ R given by f (x) = d(x, E) is continuous.

Now consider the set E of non-empty closed bounded sets in Rn with the

usual Euclidean metric d. If K, L ∈ E de¬ne

ρ (K, L) = sup{d(k, L) : k ∈ K}.

and

ρ(K, L) = ρ (K, L) + ρ (L, K).

Prove that, if K, L, M ∈ E, k ∈ K and l ∈ L then

d(k, M ) ¤ d(k, l) + ρ (L, M )

and deduce, or prove otherwise, that

ρ (K, M ) ¤ ρ (K, L) + ρ (L, M ).

Hence show that (E, ρ) is a metric space.

Suppose now that (X, d) is R with the usual metric. Suppose K is closed

and bounded in Rn (again with the usual Euclidean metric) and fn : K ’ X

is a sequence of continuous functions converging uniformly to some function

f . Show that ρ(fn (K), f (K)) ’ 0 as n ’ ∞. How far can you generalise

this result?

Exercise K.193. [11.1, T, ‘ ] The metric ρ de¬ned in the previous question

(Exercise K.192) is called the Hausdor¬ metric. It is clearly a good way of

comparing the ˜similarity™ of two sets. Its utility is greatly increased by the

observation that (E, ρ) is complete. The object of this question is to prove

this result.

(i) If Kn ∈ E and K1 ⊇ K2 ⊇ K3 ⊇ . . . explain why K = ∞ Kj ∈ E j=1

and show that ρ(Kn , K) ’ 0 as n ’ ∞.

(ii) Suppose Ln ∈ E and ρ(Ln , Lm ) ¤ 4’n for all 1 ¤ n ¤ m. If we set

¯

Kj = Lj + B(0, 2’j ) = {l + x : l ∈ Lj , x ¤ 2’j },

show that Kn ∈ E and K1 ⊇ K2 ⊇ K3 ⊇ . . . . Show also that, if we write

K = ∞ Kj we have ρ(Ln , K) ’ 0 as n ’ ∞.

j=1

(iii) Deduce that (E, ρ) is complete.

535

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

Exercise K.194. [11.2, T] Let (V, ) be a normed vector space and E a

closed subspace (that is a vector subspace which is also a closed subset). If

E = V there must exist a z ∈ V \ E. By considering

inf{ z ’ y : y ∈ E},

> 0 we can ¬nd an x ∈ V such that

or otherwise, show that given any

x = 1 but

x ’ e > 1 ’ for all e ∈ E.

Show that if V is in¬nite dimensional we can ¬nd a sequence xj ∈ V such

that

xj = 1 for all j, but xi ’ xj > 1/2 for all i = j.

¯

Deduce that if (V, ) is a normed space then the closed unit ball B =

{x : x ¤ 1} has the Bolzano-Weierstrass property if and only if V is ¬nite

dimensional.

Exercise K.195. [11.2, P] (i) Consider l ∞ the space of bounded real se-

∞ given by x ∞ = supn≥1 |xn |. Let κn be a positive

quences with norm

real number [n ≥ 1]. Show that the set

E = {x ∈ l∞ : |xn | ¤ κn for all n ≥ 1}

has the Bolzano-Weierstrass property if and only if κn ’ 0 as n ’ ∞.

(ii) Consider l1 the space of real sequences whose sum is absolutely con-

∞

n=1 |xn |. Let κn be a positive

vergent, with norm 1 given by x ∞ =

real number [n ≥ 1]. Show that the set

E = {x ∈ l∞ : 0 ¤ xn ¤ κn for all n ≥ 1}

∞

has the Bolzano-Weierstrass property if and only if κn converges.

n=1

Exercise K.196. [11.2, T] This exercise takes up ideas discussed in Exer-

cises K.29 to K.36 but the reader only needs to have looked at Exercise K.29.

We work in a metric space (X, d). Consider the following two possible

properties of (X, d).

(A) If a collection K of closed sets has the ¬nite intersection property,

then K∈K K = ….

(B) If a collection U of open sets is such that U ∈U U = X, then we can

¬nd an n ≥ 1 and U1 , U2 , . . . , Un ∈ U such that n Un = X.

j=1

(i) Show that (X, d) has property (A) if and only if it has property (B).

536 A COMPANION TO ANALYSIS

(ii) Suppose that (X, d) has the Bolzano-Weierstrass property and that

U is a collection of open sets such that U ∈U U = X. Show that there exists

a δ > 0 such that, given any x ∈ X, we can ¬nd a U ∈ U with B(x, δ) ⊆ U .

(iii) Use (ii) and Lemma 11.2.6 to show that, if (X, d) has the Bolzano-

Weierstrass property, then it has property (B).

Exercise K.197. [11.2, T, ‘ ] This exercise continues Exercise K.196.

(i) Suppose that xn ∈ X for all n but that the sequence xn has no

convergent subsequence. Show that given any x ∈ X we can ¬nd a δx > 0

and an Nx ≥ 1 such that xn ∈ B(x, δx ) for all n ≥ Nx .

/

(ii) By considering sets of the form Ux = B(x, δx ), or otherwise, show

that the space X described in (i) can not have property (B).

(iii) Deduce that a metric space (X, d) has property (B) if and only if it

has the Bolzano-Weierstrass property.

(iv) Use (iii) to prove parts (ii) and (iv) of Exercise K.36.

If (X, d) has property (B) we say that it is compact. We have shown that

compactness is identical with the Bolzano-Weierstrass property for metric

spaces. However, the de¬nition of compactness makes no explicit use of the

notions of ˜distance™ (that is, metric) or ˜convergence™ and can be generalised

to situations where these concepts cease to be useful.

Exercise K.198. [11.3, P] If f : [0, 1] ’ R is continuous we write

1

= sup |f (t)|, and f |f (t)| dt.

f =

∞ 1

t∈[0,1] 0

Consider the space C 1 ([0, 1]) of continuously di¬erentiable functions. We set

f = f ∞+ f

A 1

f =f∞

B

f = f ∞+ f ∞

C

= |f (0)| + f

f D 1

Which of these formulae de¬ne norms? Of those that are norms, which are

complete? Consider those which are norms together with the norms ∞

and 1 . Which are Lipschitz equivalent and which not? Prove all your

answers.

Exercise K.199. [11.3, T] Weierstrass™s example (see page 199) becomes

very slightly less shocking if we realise that our discussion uses an inappropri-

ate notion of when one function is close to another. In the next two exercises

we discuss a more appropriate notion. It will become clear why much of

537

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

the early work in what we now call the theory of metric spaces was done by

mathematicians interested in the calculus of variations.

(i) Find a sequence of functions fn : [’1, 1] ’ R which have continuous

derivative (with the usual convention about derivative at the end points)

such that fn ’ |x| uniformly on [’1, 1] as n ’ ∞.

(ii) If b > a let us write C 1 ([a, b]) for the space of functions on [a, b]

which have continuous derivative. Show that C 1 ([a, b]) is not closed in

1

(C([a, b]), ∞ ) and deduce that (C ([a, b]), ∞ ) is not complete.

(iii) If f ∈ C 1 ([a, b]) let us write

f =f +f ∞.

— ∞

Show that (C 1 ([a, b]), —) is a complete normed space.

(iv) Consider

A = {f ∈ C 1 ([0, 1]) : f (0) = f (1) = 0}.

Show that A is a closed subset of (C 1 ([a, b]), and that A is a vector subspace

of C 1 ([a, b]). Conclude that (A, — ) is a complete normed space. Show also

that (A, ∞ ) is not complete.

Exercise K.200. [11.3, T, ‘ ] We continue with the discussion and notation

of Exercise K.199. The example we gave on page 199 involved studying the

function I : A ’ R given by

1

(1 ’ (f (x))4 )2 + f (x)2 dx.

I(f ) =

0

(i) Show that (if we give R the usual metric) the function I : A ’ R is

not continuous if we give A the norm ∞ but is continuous if we give A

the norm —.

(ii) Do (or recall) Exercise 8.3.4 which shows that

inf{I(f ) : f ∈ A} = 0

but that I(f ) > 0 for all f ∈ A. This is the main point of the Weierstrass

counterexample and is una¬ected by our discussion.

(iii) Write f0 = 0. By using Exercise 8.4.12, or otherwise, show that we

can ¬nd fn ∈ A such that fn ’ f0 ∞ ’ 0 and Ifn ’ 0.

(iv) Show that if fn ∈ A and Ifn ’ 0 then the sequence fn does not

converge in (A, — ).

538 A COMPANION TO ANALYSIS

Exercise K.201. [11.3, T, ‘ ] Exercise K.200 shows that the question ˜Is

f0 = 0 a local minimum for I?™ should be asked using the norm — rather

than the norm ∞ but leaves the question unanswered.

(i) By considering functions of the form f (x) = sin 2πx or otherwise