Fall 2018 Problem 2: Consider the function f : [0, 1] → ℝ defined by

f(x) = \begin{cases}x log x & if x ∈ (0, 1] \\ 0 & if x = 0. \end{cases}

Be sure to justify your answer for each of the following questions.

(a) Is f Lipschitz continuous on [0, 1]?

(b) Is f uniformly continuous on [0, 1]?

(c) Suppose (p_{n}) is a sequence of polynomial functions on [0, 1], converging uniformly to f. Is the set A = {p_{n}|n ≥ 1} ∪ {f} equicontinuous?

(Spring 2015 Problem 1.) Let H be a separable Hilbert space. A sequence (x_{n}) in H converges in the Cesáro sense to x ∈ H if the averages of its partial sums converge strongly to x, i.e., if

\overline{x}_{N} = \frac{1}{N} ∑_{n=1}^{N} x_{n} → x as N → ∞.

(a) Prove that if (x_{n}) converges strongly to x in H, then (x_{n}) also converges in the Cesáro sense to x.

(b) Give an example of a sequence that converges in the Cesáro sense but does not converge weakly.

(c) Give an example of a sequence that converges weakly but does not converge in the Cesáro sense.

(Spring 2015 Problem 4) Let f_{n} : [0, 1] → ℝ be a sequence of measurable functions. Suppose

(i) ∫_{0}^{1} |f_{n}(x)|^{2}dx ≤ 1 for n = 1, 2, · · ·.

(ii) f_{n} → 0 almost everywhere.

Show that

lim_{n→∞} ∫_{0}^{1} f_{n}(x)dx = 0.

(Fall 2015 Problem 1.) Consider the following sequences of functions parameterized by n:

• a_{n}(x) = e^{i2\sqrt{n}πx}, x ∈ [0, 1],

• b_{n}(x) = \sqrt{n}e^{−n|x|}, x ∈ ℝ,
• c_{n}(x) = ne^{−nx^{2}}, x ∈ ℝ,
• d_{n}(x) = ∑_{k=−n}^{n} e^{i2kπx}, x ∈ [0, 1].

As n tends to infinity, which sequences converge (a) almost everywhere, (b) L^{2}- strongly, (c) L^{2}-weakly but not strongly. Explain your answer.

(Fall 2014 Problem 3) Let H be a separable Hilbert space with inner product ⟨·, ·⟩ : H × H → ℂ, and let {e_{n} : n ∈ ℕ} be an orthonormal basis of H. Define a metric d : B × B → ℝ on the closed unit ball B of H by

d(x, y) = ∑_{n=1}^{∞} \frac{|⟨x − y, e_{n}⟩|}{2^{n}}.

(a) Show that the sequence (x_{k}) in B converges to x ∈ B with respect to the metric d if and only if it converges weakly to x in H.

(b) Prove that (B, d) is a compact metric space.

(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.