Carnegie Mellon University Department of Mathematical Sciences
Functional Analysis Basic Exam – Fall 2018
Closed-book and closed-notes, three hours
Answer questions 1 and 2 plus any other six questions (8 total). If you
answer all of the last 7 questions, I will use your best 6 for computing
your grade. (You have nothing to lose by attempting all questions.) Be sure to give
complete and clear explanations in your answers to questions 3 thru 9.
1. (20 pts) Let l2denote the set of all real-valued square-summable sequences,
equipped with the usual norm
kxk= 1
X
k=1
(xk)2!
1
2
for all x2l2.
Define T:l2!l2by
(Tx)k=⇢0ifkis odd
xk
2if kis even for all x2l2,
i.e.
Tx =(0,x
1,0,x
2,0,x
3,···).
You m ay take i t for gr a nted t h a t Tis a bounded linear operator from l2to l2.
(a) Is Tinjective? (Give a brief explanation.)
(b) Is R(T)denseinl2?(Giveabriefexplanation.)
(c) Assuming that we identify the dual space (l2)⇤with l2in the usual way,
find an expression for the adjoint operator T⇤.
(d) Is Tcompact? Explain.
(e) Give an example of a bounded linear mapping L:l2!l2such that
TL 6=LT .
2. (20 pts)
(a) State the Hahn-Banach Theorem for linear spaces in geometric form (i.e.,
state a result about separation of convex sets).
(b) State the Banach-Steinhaus Theorem (also called the Principle of Uniform
Boundedness).
(c) State the Closed Graph Theorem.
(d) State the Riesz-Representation Theorem for Hilbert spaces.
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