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(c) the Fj have bounded diameters but the diameters do not tend to 0.

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

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