Department of Mathematics
MATH4010 Functional Analysis
Homework 3
Notice:
• All the assignments must be submitted before the deadline.
• Each assignment should include your name and student ID number.
1. Let p ∈ (0, 1). Define
`p
:=
(
(xk)
∞
k=1 ∈ C:
X∞
k=1
|xk|
p < ∞
)
.
For x = (xk)
∞
k=1 and y = (yk)
∞
k=1 in `p, define the metric d by
d(x, y) = X∞
k=1
|xk − yk|
p
.
Then (`p, d) is a metric vector space. Let (bk)
∞
k=1 be a bounded sequence in C. Show that
f(x) = X∞
k=1
bkxk for x = (xk)
∞
k=1 ∈ `p
is a continuous linear functional on the metric vector space (`p, d).
2. Let C[0, 1] be the vector space of continuous functions on [0, 1]. Define δ(x) = x(0) for
x ∈ C[0, 1].
(a) Show that δ is a bounded linear functional if C[0, 1] is endowed with the sup-norm. Find
the norm of δ.
(b) Show that δ is an unbounded linear functional if C[0, 1] is endowed with the norm
kxk =
Z 1
0
|x(t)|dt.
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