Spring 2014 Problem 3 Let f, f_{k} : E → [0, +∞) be nonnegative Lebesgue integrable functions on a measurable set E ⊆ ℝ^{n}. If (f_{k}) converges to f pointwise almost everywhere and

∫_{E} f_{k}dx → ∫_{E} fdx,

show that

∫_{E} |f − f_{k}|dx → 0.

Fall 2017 Problem 3 Prove or disprove the following statement: If f ∈ C^{∞}([0, 1]) is a smooth function, then there is sequence (p_{n}) of polynomials on [0, 1] such that p_{n}^{(k)} → f^{(k)} uniformly on [0, 1] as n → ∞ for every integer k ≥ 0. Here f^{(k)} denotes the kth derivative of f.

Fall 2015 Problem 6 Suppose that f ∈ S(ℝ), where S(ℝ) is the Schwartz space of infinitely differentiable rapidly decreasing functions

S(ℝ) = {f ∈ C^{∞}(ℝ) : sup_{x∈ℝ} |x^{n}f^{(m)}(x)| < ∞}

for all nonnegative integers

n, m = 0, 1, 2, ….

Does

∫ℝ f(x)x^{n}dx = 0, n = 0, 1, 2, ….

imply that f is identically zero? Explain your answer. (Hint: use Fourier transform).

Note: f^{(m)} denotes the m-th derivative of f.

Fall 2013 Problem 2 Let L^{2}([0, 1]) denote the Hilbert space of complex valued square integrable functions on [0, 1] with the usual inner product

(f, g) = ∫_{0}^{1} f(x)\overline{g(x)}dx.

Define T : L^{2}([0, 1]) → L^{2}([0, 1]) by

(T f)(x) = ∫_{0}^{x} f(t)dt, for x ∈ [0, 1].

(a) Show that T is bounded.

(b) Show that T has no eigenvalues.

(c) Find lim_{n→∞} ∥T^{n}∥.

Spring 2014 Problem 2 Let T be a linear operator from a Banach space X to a Hilbert space H. Show that T is bounded if and only if x_{n} ⇀ x implies that T (x_{n}) ⇀ T (x) for every weakly convergent sequence (x_{n}) in X.

Spring 2015 Problem 6 Let {A_{n}} be a sequence of bounded linear operators on a Hilbert space H that converges weakly to an operator A, and suppose that for each x ∈ H one has ∥A_{n}x∥ → ∥Ax∥ as n → ∞. Prove that A_{n} strongly converges to A (in particular, for unitary operators weak convergence to a unitary operator implies strong convergence).

Fall 2016 Problem 1: Let A and B be two bounded, self-adjoint operators on a Hilbert space H. Prove that

∥Af∥∥Bf∥ ≥ \frac{|⟨[A, B]f, f⟩|}{2},

where [A, B] = AB − BA is the commutator of A and B. In addition prove that equality holds if and only if Af = cBf for some c ∈ ℝ.