0 there exists > 0
such that kxk = kyk = 1 and k1
2 (x + y)k > 1 imply that kx yk < “.
Prove that any Hilbert space is uniformly convex.
You may use without proof the parallelogram law
kx + yk2
+ kx yk2
= 2kxk2
+ 2kyk2
:
[11 marks]
2. A Let 1 p < 1. State precisely what it means to say that a function
f : [0; 1] ! C belongs to Lp
([0; 1]). Dene the (usual) norm on this space.
[3 marks]
B Does there exist a function f 2 L1
(R) such that limsupx!1 f(x) = 1
Justify brie y your answer. [11 marks]
C Give an example of a set A L2
([0; 1]), which is unbounded in L2
([0; 1]),
but bounded in L1
([0; 1]). [11 marks]
3. A (i) State the axioms of an inner product on a complex linear space.
(ii) State (but do not prove) the Riesz representation theorem in a Hilbert
space. [3 marks]
B Let M be a closed subspace of a Hilbert space H and suppose that T : M !
C is a bounded linear functional. Show that there exists a b
ounded linear
functional S : H ! C such that S = T on M and kSk = kTk.
You may use any result about Hilbert spaces from the course but you may
not use the Hahn-Banach theorem. [8 marks]
C Let X be a Hilbert space. Prove that the following assertions are true or
provide counterexamples showing that they are false:
(i) If M is a dense linear subspace of X, then M = f0g.
(ii) If M is a linear subspace of X such that M = f0g, then M is dense.
Here M denotes the orthogonal complement of M in X. [14 marks]
4. A Let X be a Banach space.
(i) Explain what it means to say that : X ! C belongs to the dual space
X
. Write down the denition of the norm of 2 X
.
(ii) What does it mean to say that X is re exive [3 marks]
B Let 1 < p < 1 and dene : Lp
[ 1; 1] ! C by
f !
Z 1
1
f(x)dx:
Prove that is well-dened bounded linear map and compute its norm.
You may use Holder’s inequality but no other results from the course.