MATH 5410, Preliminary Exam
Department of Mathematics
University of Connecticut
January, 2011
NAME: SIGNATURE:
1. a) What is the definition of a self-adjoint operator from a Hilbert space Hto itself;
b) Give an example of a self-adjoint operator for H=L2([0,1]) and explain;
c) Prove that eigenvalues of a self-adjoint operator must be real.
2. a) What is the definition of weak convergence of a sequence {xn}in a Hilbert space H;
b) Prove that a strongly convergent sequence is also a weakly convergent sequence in H;
c) Give an example of a weakly convergent sequence which is NOT strongly convergent in l2and explain;
3. a) Give the definition of the limit of a sequence of distributions {fn}∞
1in Ras n→ ∞.
b) Let
f(x) = e−x2, fn(x) = nf(nx),∀x∈R, n = 1,2,·· ·.
How do you interpret function fn(x) as a distribution fnin R?
c) Find the limit of {fn}∞
1as a sequence of distributions as n→ ∞.
(You may use the fact that RRe−x2dx =√π).
4. a) Suppose fis an operator from Banach space Xto itself. Give the definition of fbeing Fr´echet
differentiable at a point x∈X.
b) Let X=C[0,1] with sup-norm. Let ti∈[0,1] and vi∈C[0,1], and define f(x) = Σn
i=1(x(ti)2)vi.
Prove that fis Fr´echet differentiable at all points of Xand find a formula for f0.
5. Find a function in C1[0,1] that minimizes the integral R1
0[(u0(t))2+u2(t)]dt with constraints u(0) = 0
and u0(1) = 1.
6. Let [un] be an orthonormal sequence in a Hilbert space and let [λn] be a bounded sequence in R. Prove
that the operator Ax = Σλn< x, un> unis compact if and only if λn→0 as n→ ∞.
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