MATH 421/510
HW 2
Due: February 11, 2016 (in class)
1. Let X be a reflexive normed vector space and let T ∈ X
∗
. Show that there exists x
0
∈ X such that k x
0 k = 1 and k T k := sup x ∈ X, k x k =1 k T x k = T x
0
.
2. Let X denote the space of continous functions on [0 , 1] equipped with sup norm. Recall that X is a Banach space.
(a) Show that
T f =
Z
1 / 2 f ( x ) dx −
0
Z
1
1 / 2 f ( x ) dx is a bounded linear functional.
(b) Compute k T k .
(c) Does there exist f
0
∈ X such that k f
0 k = 1 and k T k = T f
0
?
(d) Is X reflexive?
3. Let X be a normed vector space and let X
∗ a subspace M of X , we set denote the dual space. For
M
⊥
:= { f ∈ X
∗
: f ( x ) = 0 ∀ x ∈ M } .
For a subspace N of X
∗
, we define
N
⊥
:= { x ∈ X : f ( x ) = 0 ∀ f ∈ N } .
Prove that M
⊥
⊥
= M .
4. Chapter 5, Exercise 59.
5. Consider the set
Z
1
C = f ∈ L
1
([0 , 1]) : f ( t ) dt = 1 .
0
Show that C is a closed convex subset of L
1
([0 , 1]) which contains infinitely many elements of minimal norm. (Compare it to the previous question).