Functional Analysis, Spring 2015
Homework # 1
due on Wednesday January 28 in class
1. Show that a normed linear space X is a Banach space
Pif and only if
for every sequence (xn ) of vectors
in
X,
the
condition
n=1 kxn k <
P
implies the convergence of n=1
Functional Analysis, Spring 2015
Homework #9
This assignment is due on Wednesday, April 8
1. Let H be an infinite dimensional Hilbert space. Show that there
is T L(H) such that T 2 is compact but T is not compact.
2. Define T : `2 (N) `2 (N) by
T (x0 , x1
Functional Analysis, Spring 2015
Homework #8
This assignment is due on Wednesday, March 25
We make the following blanket assumption. Let H be a Hilbert space
and assume that T L(H) is a normal operator.
1. Show that Ker(T ) = Ker(T n ) for any n 1.
2. If
Functional Analysis, Spring 2015
Homework #7
This assignment is due on Wednesday, March 11
(1) Let H be a Hilbert space and assume that T L(H) is normal,
i.e. T T = T T . If (T ) = cfw_1 , . . . , n contains n distinct points,
show that T = 1 P1 + + n Pn
Functional Analysis, Spring 2015
Homework #6
This assignment is due on Wednesday, March 4
1) Let X be a compact Hausdorff space and let C(X) be the Banach
algebra of all complex valued continuous functions on X. Show that
every maximal proper ideal of C(X
Functional Analysis, Spring 2015
Homework #5
This assignment is due on Wednesday, February 25
1) Let H be the Hilbert space L2 ([0, 1], dt). Define T L(H) by
(T x)(t) = tx(t) for all t [0, 1] and x H.
(a) Show that the spectrum of T , L(H) (T ) = [0, 1].
Functional Analysis, Spring 2015
Homework #4
This assignment is due on Wednesday, February 18
1. Let p = p2 L(H) where H is a Hilbert space. Compute the
spectrum of p.
2. Define T : L2 [0, 1] L2 [0, 1] by
Z x
(T f )(x) = f (x) +
yf (y)dy,
f L2 [0, 1], x [
Functional Analysis, Spring 2015
Homework #3
This assignment is due on Wednesday, February 11 in class
1. Prove that (a) holds in a real Hilbert space, (b) in a complex
Hilbert space and (c) in any Hilbert space:
1
(a) hx, yi = (kx + yk2 kx yk2 ).
4
1
(b)
Functional Analysis, Fall 2015
Homework # 2
due on Wednesday February 5 in class
1. Show that any finite dimensional linear subspace of a normed
linear space is closed.
2. Let X be an infinite dimensional Banach space. Show that any
vector space basis of
Functional Analysis, Spring 2015
Homework #10
This assignment is due on Wednesday, April 15
1). Let H be a Hilbert space and let T L(H) be a Fredholm
operator such that T T = T T . Show that index(T ) = 0.
2) Let S : `2 (N) `2 (N) be the unilateral shift: