MATH 421/510 HW 2 Due: February 11, 2016 (in class)

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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).

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