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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