REAL ANALYSIS III - FINAL EXAM
Spring 2015
Instructions: Solve the problems below. Write your solutions – each on a separate sheet of paper clearly marking
the problem number. Staple the solutions together with this front page, sign your exam and bring it to my office
or submit by email in the electronic form no later then 8PM on Tuesday, June 9th.
Print Name:
Signature:
Problem score/max Problem score/max
1
/ 10
3
/ 10
2
/ 10
4
/ 10
Total:
/ 40.
Problem 1. Prove that a bounded set in C 1 ([0, 1]) (i.e. kf kC 1 = kf k∞ + kf 0 k∞ is bounded) is relatively
compact in L∞ ([0, 1]).
Problem 2. Let 1 ≤ p < ∞. Consider the operator A : D(A) ⊂ lp → lp given by A(xk ) = (kxk ), with the
domain of A: D(A) = {(xk ) ∈ lp : (kxk ) ∈ lp }.
(a) Prove that D(A) is a dense subspace of lp ;
(b) Show that A is linear and bijective, but not bounded (i.e. images of bounded subsets from D(A) are
not necessarily bounded).
(c) Show that the graph of A is closed, and use it to prove that the of the inverse of A viewed as an operator
A−1 : lp → lp is a bounded linear operator.
Problem 3. Consider the Banach space C([0, 1]) equipped with with the sup-norm.
(a) Show that the closed unit ball BC([0,1]) is not weakly compact on C[0, 1]. (Hint: Let Uα = {f ∈ C[0, 1] :
f (α) > 1/3}, 0 < α < 1/2 and Vβ = {f ∈ C[0, 1] : f (β) < 2/3}, 1/2 < β < 1. Then Uα , Vβ are weakly
open, cover BC([0,1]) , yet have no finite sub-cover.)
(b) Conclude that C([0, 1]) is not reflexive.
Problem 4. Let H be a Hilbert space and let E = {en }n∈N be an orthonormal family in H.
(a) Denote by H∞ the closure in H of a subspace formed by finite linear combinations of vectors in E:
∞
P
H∞ = span{e1 , . . . , en , . . . }. Prove that for any u ∈ H, the projection PH∞ (u) =
(u, en ) en .
(b) Prove that for any u ∈ H,
∞
P
n=1
2
2
|(u, en )| ≤ kuk .
n=1
(c) Prove that E is an orthonormal basis for H if and only if the expression in (b) holds with equality.
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