Spring 2014 Problem 4:

Let P_{1} and P_{2} be a pair of orthogonal projections onto H_{1} and H_{2}, respectively, where H_{1} and H_{2} are closed subspaces of a Hilbert space H. Prove that P_{1}P_{2} is an orthogonal projection if and only if P_{1} and P_{2} commute. In that case, prove that P_{1}P_{2} is the orthogonal projection onto H_{1} ∩ H_{2}.

Fall 2017 Problem 6: Consider a linear functional ϕ(f) = f(\frac{1}{2}) defined on the space of polynomials on [0, 1]. Does ϕ extend to a bounded linear functional on L^{2}([0, 1])? Prove or disprove.

Fall 2013 Problem 1: Find inf_{f} ∫_{0}^{1} |f(x)−x|^{2}dx where the infimum is taken over all f ∈ L^{2}([0, 1]) such

that ∫_{0}^{1} f(x)(x^{2} − 1)dx = 1.

Spring 2016 Problem 6: Let H be a Hilbert space and let U be a unitary operator, that is surjective and

isometric, on H. Let I = {v ∈ H : Uv = v} be the subspace of invariant vectors with respect to U.

(1) Show that {Uw − w : w ∈ H} is dense in I^{⊥} and that I is closed.

(2) Let P be the orthogonal projection onto I. Show that

\frac{1}{N} ∑_{n=1}^{N} U^{n}v → P v.

Fall 2016 Problem 3: Let {a_{n}}_{n=1}^{∞} be a sequence of real valued scalars and assume that the series

∑_{n=1}^{∞} a_{n}x_{n} converges for every x ∈ ℓ^{2}(ℕ). Show that y ∈ ℓ^{2}(ℕ).

Fall 2016 Problem 6:

Let D denote the closed unit disk in ℂ, and consider the complex Hilbert space

H := {f : D → ℂ|f(z) = ∑_{k=1}^{∞} a_{k}z^{k} and ∥f∥H^{2} := ∑_{k=0}^{∞}(1 + k^{2})|a_{n}|^{2} < ∞} .

Prove that the linear functional L : D → ℂ defined by L(f) = f(1) is bounded, and find an element g ∈ H such that L(f) = ⟨g, f⟩. (In other words so that g represents L as in the Riesz representation theorem.)