Department of Mathematics
MATH4010 Functional Analysis
Homework 2
Notice:
• All the assignments must be submitted before the deadline.
• Each assignment should include your name and student ID number.
1. Show that vectors (en), where en is the sequence whose n-th term is 1 and all other terms are
zero,
e1 = (1, 0, 0, . . .),
e2 = (0, 1, 0, . . .),
· · ·
form a Schauder basis in `
p
for every p ∈ [1, +∞) and in the spaces c0 and c00.
2. Let X = {x ∈ C[0, 1]: x(0) = 0} with the sup-norm, and let f be a linear functional on X
defined by
f(x) = Z 1
0
x(t)dt.
Show that kfk = 1.
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