Department of Mathematical Sciences
Carnegie Mellon University
Functional Analysis
Sample Exam
Do any 4 of the following 6 problems. All problems carry equal weight.
1. (a) Let Xbe a Banach space, Ybe a normed linear space, and {Tn}โ
n=1 be a
sequence of bounded linear mappings from Xto Ysatisfying
โxโX, sup{kTnxk:nโN}<โ.
Prove that
sup{kTnk:nโN}<โ.
(Do not simply quote the Banach-Steinhaus Theorem (aka the Principle
of Uniform Boundedness). You are being asked to prove that theorem.)
(b) Give an example of a normed linear space X, a Banach space Y, and a
sequence {Tn}โ
n=1 of bounded linear mappings from Xto Ysatisfying
โxโX, sup{kTnxk:nโN}<โ,
but
sup{kTnk:nโN}=โ.
2. (a) Let Xbe a reflexive Banach space and let {Kn}โ
n=1 be a sequence of
bounded subsets of Xsatisfying (i) and (ii) below.
(i) โnโN, Kn6=โ
, Knis closed, Knis convex,
(ii) โnโN, Kn+1 โKn.
Show that
โ
\
n=1
Kn6=โ
.
(b) Give an example of a (nonreflexive) Banach space Xand a sequence
{Kn}โ
n=1 of bounded subsets of Xsatisfying (i) and (ii) above, but with
โ
\
n=1
Kn=โ
.
3. Let Xbe a complex Hilbert space with inner product (ยท,ยท) and Abe a bounded
linear mapping from Xto X. Prove that Ais compact if and only if (Axn, xn)โ
0 as nโ โ for every sequence {xn}โ
n=1 such that xnโ0 (weakly) as nโ โ.
What happens with regard to this result in real Hilbert spaces?
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