MATH 5410, Preliminary Exam
Department of Mathematics
University of Connecticut
August 23, 2010
NAME: SIGNATURE:
1. a) What is the definition of a compact linear operator from a Banach space Xto itself;
b) Give an example of an operator for X=L2([0,1]) which is a compact linear operator and explain
why;
c) Give an example of an operator for X=L2([0,1]) which is NOT a compact linear operator and explain
why;
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 L2([0,1]) and
explain;
3. a) Give an example of a distribution which can NOT be identified with a continuous function in Rand
explain why.
b) Define δ(0) as a distribution;
c) If T(φ) = φ(0) + φ0(1) for every φ∈ D(R), find ∂T the derivative of T.
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))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+u(t)]dt with constraints u0(0) = 0
and u(1) = 1.
6. Find an orthonormal basis for L2[0,1] by considering the Sturm-Liouville operator
Ax =x00 +xwith x(0) = x(1) = 0. Explain the reasons (theory) behind your method.
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