querent

Spam yourself with math PhD qualifying-exam questions.

UC Irvine Fall 2023, 2 Real analysis, absolute continuity, composition of functions, monotonicity

Suppose that $𝑓 : ℝ \to ℝ$ is absolutely continuous and $𝑔 : [𝑎, 𝑏] \to ℝ$ is absolutely continuous and strictly monotonic. Prove that $𝑓 \circ 𝑔 : [𝑎, 𝑏] \to ℝ$ is absolutely continuous.

Stanford Fall 2023, 3 Real analysis, bounded operators, Banach spaces, strong operator topology

Let $𝑋$ be a Banach space and $ℐ︀ \subset ℒ︀ (𝑋, 𝑋)$ be the set of invertible bounded linear operators.

(a) Prove that $ℐ︀ \subset ℒ︀ (𝑋, 𝑋)$ is open with respect to the operator norm topology.

(b) Is $ℐ︀ \subset ℒ︀ (𝑋, 𝑋)$ necessarily open with respect to the strong operator topology? Prove this or give a counterexample.

Stanford Fall 2023, 3 Real analysis, Laplace transform, Lp spaces, bounded operators

Let $ℒ︀ 𝑓$ denote the Laplace transform

ℒ︀ 𝑓 (𝑠) = \int_{0}^{\infty} 𝑒^{- 𝑥 𝑠} 𝑓 (𝑥) d 𝑥 .

Prove that $ℒ︀$ is a bounded operator on $𝐿^{𝑝 ([0, + \infty))}$ if and only if $𝑝 = 2$ .

UC Irvine Fall 2023, 3 Real analysis, weak derivatives, constant functions, L^infinity functions

Suppose that $𝑓 \in 𝐿^{\infty} (0, 1)$ is such that

\int_{(0, 1)} 𝑔^{'} (𝑥) 𝑓 (𝑥) d 𝑥 = 0

for every absolutely continuous function $𝑔$ on $[0, 1]$ for which $𝑔 (0) = 𝑔 (1) = 0$ . Prove that $𝑓$ equals to a constant almost everywhere.

Stanford Fall 2023, 4 Real analysis, Sobolev spaces, Poincaré inequality

(a) Prove that there is a constant $𝐶_{1} > 0$ such that $\int_{0}^{1} 𝑤^{2} (𝑡) d 𝑡 \leq 𝐶_{1} \int_{0}^{1} {(𝑤^{'})}^{2} (𝑡) d 𝑡$ for all $𝑤 \in 𝐶^{\infty} ([0, 1])$ satisfying $𝑤 (0) = 0 = 𝑤 (1)$ .

(b) Prove that there is a constant $𝐶_{2} > 0$ such that

\int_{- \infty}^{\infty} 𝑢^{6} (𝑡) d 𝑡 \leq 𝐶_{2} \int_{- \infty}^{\infty} {(𝑢^{'})}^{2} (𝑡) d 𝑡

for all $𝑢 \in 𝐶_{𝑐}^{\infty} (ℝ)$ satisfying $\int_{- \infty}^{\infty} 𝑢^{2} (𝑡) d 𝑡 = 1$ . (Hint: Justify the change of variables $𝑠 = \int_{- \infty}^{𝑡} 𝑢^{2} (𝑦) d 𝑦$ .)

Stanford Fall 2023, 4 Real analysis, uniform boundedness, Lp spaces

Suppose $𝐻$ is a closed subspace of $𝐿^{2} ([0, 1])$ such that $𝐻 \subset 𝐶 ([0, 1])$ . Prove that $𝐻$ is finite-dimensional.

UC Irvine Fall 2023, 4 Real analysis, Young's inequality, convolution, continuity

Fix $𝑓 \in 𝐿^{\frac{6}{5}} (ℝ)$ and $𝑔 \in 𝐿^{6} (ℝ)$ . Let $ℎ$ be the function on $ℝ$ defined by

ℎ (𝑥) = \int_{- \infty}^{\infty} 𝑓 (𝑥 - 𝑦) 𝑔 (𝑦) d 𝑦 .

Prove that $ℎ$ is continuous.

Stanford Fall 2023, 5 Real analysis, cluster points, countable sets

For $𝐸 \subset ℝ^{𝑛}$ and $𝑓 : 𝐸 \to ℝ^{𝑛}$ , let

𝐹 = {𝑥 \in 𝐸 : \exists {𝑥_{𝑘}}_{𝑘 = 1}^{\infty} \subset 𝐸 ∖ {𝑥} such that 𝑥_{𝑘} \to 𝑥 and 𝑓 (𝑥_{𝑘}) \to 𝑓 (𝑥)} .

Prove that $𝐸 ∖ 𝐹$ is at most countable.

Stanford Fall 2023, 5 Real analysis, Hilbert space geometry, Volterra operator, Hilbert-Schmidt operators

Let $𝐻$ be a separable Hilbert space and ${𝑒_{𝑗}}_{𝑗 = 1}^{\infty}$ be an orthonormal basis. A bounded operator $𝐴$ on $𝐻$ is called Hilbert–Schmidt if

\sum_{𝑗 = 1}^{\infty} {‖ 𝐴 𝑒_{𝑗} ‖}^{2} < \infty . (1)

(a) Show that (1) implies that $\sum_{𝑗 = 1}^{\infty} {‖ 𝐴 𝑒_{𝑗}^{'} ‖}^{2} < \infty$ for any orthonormal basis ${𝑒_{𝑗}^{'}}_{𝑗 = 1}^{\infty}$ of $𝐻$ .

(b) Prove that the set of Hilbert–Schmidt operators on $𝐻$ is itself a Hilbert space, with inner product .

𝑉 𝑢 (𝑥) = \int_{0}^{𝑥} 𝑢 (𝑦) d 𝑦, 𝑥 \in [0, 1],

is Hilbert–Schmidt.

UC Irvine Fall 2023, 5 Real analysis, L^p spaces, weighted integral bounds

Fix $1 \leq 𝑝 \leq \infty$ and $𝑓 \in 𝐿^{𝑝} (ℝ)$ . Show that

\int_{- \infty}^{\infty} | 𝑓 (𝑡) \frac{|}{1 + 𝑡^{2}} d 𝑡 < \infty .

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