Department of Mathematics
MATH4010 Functional Analysis
Homework 5
Notice:
• All the assignments must be submitted before the deadline.
• Each assignment should include your name and student ID number.
1. Let (xn) be a sequence in an inner product space. Show that the conditions kxnk → kxk and
hxn, xi → hx, xi imply xn → x.
2. Show that
X =
n
x = (xn) ∈ `
2
:
X∞
n=1
xn
n
= 0o
is a closed subspace of `
2
.
3. (a) Prove that for every two subspaces X1 and X2 of a Hilbert space,
(X1 + X2)
⊥ = X
⊥
1 ∩ X
⊥
2
.
(b) Prove that for every two closed subspaces X1 and X2 of a Hilbert space,
(X1 ∩ X2)
⊥ = X⊥
1 + X⊥
2
.
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