Math 310 Preliminary Exam
Fall 2008
Name
Signature
1. (20 pts)
(a) Let 02(0;1) :Find the Green’s Function for
y00 +y=(t0)
y0(0) = y(1) = 0:
(b) Show that there exists a unique solution for
y00 +y=tan1y+ cos x
y0(0) = y(1) = 0
for jjsu¢ ciently small.
2. (20 pts) Let Tbe a compact operator on a Hilbert space Hand f'n:n2Ng
be an orthonormal system of H:
(a) (5 pts) Show 'n*0weakly. Explain why this gives an example of
weakly convergent sequence which is not strongly convergent.
(b) (5 pts) Using a. or otherwise, show kT'nk ! 0
(c) (10 pts) Let nbe a sequence of complex numbers. Then operator S
de…ned by Sf =P1
n=1 nhf; 'ni'nis compact i¤ limn!1 n= 0:
3. (20 pts)
(a) Let fbe an operator on a Banach space X; give the de…nition of f
being Fréchet di¤erentiable at a point x2X:
(b) De…ne f:C[0;1] ! C[0;1] by [f(x)] (t) = x(t) + R1
0(x(st))2ds:
Compute f0(x):
4. (20 pts) Let Kbe a compact operator on a Banach space. If I+Kis
injective, then it is surjective.
5. (20 pts) Let T(') = '(1) + '(1) for every '2 D (R):
(a) Show Tis a distribution.
(b) Find @T the distributional derivative of T:
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