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Name:
Math 5410 Prelim August , 2011
Choose 5 out of the 6 questions.
(1a) Let H be a real Hilbert space with inner product 〈·, ·〉. Establish the parallelogram law, i.e., for all x, y ∈ H, one has:
‖x + y‖^2 + ‖x − y‖^2 = 2‖x‖^2 + 2‖y‖^2.
(1b) Let K be a non-empty closed convex set in H. Show that for any x ∈ H, there exists a unique point y ∈ K such that
‖x − y‖ = dist(x, K).
(hint: use (a) to show a sequence {yn} that attains dist(x, K) is a Cauchy sequence).
(1c) Let x ∈ X and let y be the point of K closest to x as in (b). Prove that 〈x − y, v − y〉 ≤ 0 for all v ∈ K.
(2a) Find the Green’s function G(x, y) for the operator A where
Au = −u′′^ + u
with u(0) = u′(1) = 0. (2b) Define T : L^2 (0, 1) → L^2 (0, 1) such that for any f ∈ L^2 (0, 1),
(T f )(x) =
∫ (^1)
0
G(x, y)f (y) dy.
Explain what spectral theorem is and why it is applicable. (2c) Show that ‖T ‖ = max{|λ| : λ is an eigenvalue of T}. (2d) Compute ‖T ‖. (hint: find eigenvalues of A).
(3) Let k > 0 and u : R^3 \ { 0 } → R defined by
u(x) ≡ −
4 π|x|
e−k|x|^.
It is spherically symmetric, i.e. u(x) = w(|x|) where w : R → R is given by w(r) = − (^4) πr^1 e−kr.
(a) For a spherical symmetric function in R^3 , it is known that
∆u = [
d^2 dr^2
r
d dr
]w.
Show that (∆ − k^2 )u = 0 for x 6 = 0.
(b) Show that the distribution ˜u is a fundamental solution of the Helmholtz operator ∆ − k^2 , i.e. (∆ − k^2 )˜u = δ.
(hint: the proof is similar to that for k = 0 when we deal with the Laplacian operator).
(4) Let H be a Hilbert space and A : H → H is compact. (a) Give the definition that A is compact.
(b) Let {un} be an orthonormal sequence in a Hilbert space and let {λn} be a bounded sequence in R. Prove that the operator Ax =
∑ λn〈x, un〉un is compact if and only if λn → 0 as n → ∞.
