## Description

Department of Mathematics

MATH4010 Functional Analysis

Homework 4

Notice:

• All the assignments must be submitted before the deadline.

• Each assignment should include your name and student ID number.

1. If X and Y are Banach spaces and Tn : X → Y, n = 1, 2, . . . a sequence of bounded linear

operators, show that the following statements are equivalent:

(a) the sequence (kTnk) is bounded,

(b) the sequence (kTnxk) is bounded for every x ∈ X,

(c) the sequence (|f(Tnx)|) is bounded for every x ∈ X and every f ∈ Y

∗

.

2. Show that the space

Y = {X ∈ C

1

[0, 1]: x(0) = 0}

equipped with the sup-norm is not a Banach space (cf. the following lemma).

Lemma. The sequence

xn(t) = r

(t −

1

2

)

2 +

1

n

, t ∈ [0, 1]

converges uniformly to the function x(t) = |t − 1/2| on [0, 1].

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