querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Fall 2025, 2 Real analysis, Lp spaces, Lebesgue measure, concentration inequalities

Suppose that $𝑓 \in 𝐿^{2} ([0, 1])$ , $\int_{0}^{1} 𝑓 = 0$ , and $\int_{0}^{1} {| 𝑓 |}^{2} > 0$ .

(a) Let $𝑎_{𝑛}$ be a sequence of positive numbers that converges to $\infty$ . Consider

𝐹_{𝑛} = {(𝑥_{1}, \dots, 𝑥_{𝑛}) \in {[0, 1]}^{𝑛} : | \sum_{𝑗 = 1}^{𝑛} 𝑓 (𝑥_{𝑗}) | \geq 𝑎_{𝑛} 𝑛^{1 / 2}} .

Prove that ${lim}_{𝑛 \to \infty} 𝑚_{𝑛 (𝐹_{𝑛})} = 0$ , where $𝑚_{𝑛}$ is the Lebesgue measure on $ℝ^{𝑛}$ .

(b) Suppose that $𝑎_{𝑛} = 1$ , prove that ${lim}_{𝑛 \to \infty} 𝑚_{𝑛 (𝐹_{𝑛})}$ exists and is not equal to zero.

Stanford Fall 2025, 2 Real analysis, Schwartz functions, distributions, commuting operators

Let $𝑀 : 𝒮︀ (ℝ) \to 𝒮︀ (ℝ)$ denote the map $𝑀 𝑓 (𝑥) = 𝑥 𝑓 (𝑥)$ . Suppose $𝑇 : 𝒮︀ (ℝ) \to 𝒮︀ (ℝ)$ is a linear map (with no boundedness assumed) satisfying $𝑇 𝑀 = 𝑀 𝑇$ and $𝑇 \frac{d}{d 𝑥} = \frac{d}{d 𝑥} 𝑇$ . Prove that there exists $𝑐 \in ℂ$ such that $𝑇 𝑓 = 𝑐 𝑓$ for all $𝒮︀ (ℝ)$ . (Hint: First show that if $𝑦 \in ℝ$ and $𝑓 (𝑦) = 0$ , then $(𝑇 𝑓) (𝑦) = 0$ . This then implies that $𝑇$ is multiplication by a function.)

UC Irvine Fall 2025, 2 Real analysis, limits at infinity, derivatives

Suppose $𝑓 : (0, \infty) \to ℝ$ is continuously differentiable and

lim_{𝑥 \to \infty} 𝑓^{'} (𝑥) = 𝐿 \in ℝ .

Define

𝑔 (𝑥) = 𝑓 (𝑥 + 1) - 𝑓 (𝑥) .

Prove that

lim_{𝑥 \to \infty} 𝑔 (𝑥) = 𝐿 .

UC Irvine Fall 2025, 2 Real analysis, Lebesgue integrability, iterated integrals, convergence criteria

Let

Ω = {(𝑥_{1}, 𝑥_{2}) \in ℝ^{2} : 0 < 𝑥_{2} < 𝑥_{1} \leq 1} .

Let

𝑓 (𝑥) = {(𝑥_{1}^{2} + 𝑥_{2}^{2})}^{- 𝑏 / 2} | log (𝑥_{1}^{2} + 𝑥_{2}^{2}) |^{- 𝛾} .

Determine for which values of $𝑏 \in ℝ$ and $𝛾 \in ℝ$ the Lebesgue integral

\int_{Ω} 𝑓 (𝑥) d 𝑥

is finite.

Stanford Fall 2025, 3 Real analysis, weak-star topology, dual spaces, norm equivalence

For $𝑥 \in [0, 1]$ , define $𝛿_{𝑥} \in {(𝐶 ([0, 1]))}^{*}$ by $𝛿_{𝑥 (𝑓)} = 𝑓 (𝑥)$ .

(a) Prove that the map $𝑥 \mapsto 𝛿_{𝑥}$ is continuous for ${(𝐶 ([0, 1]))}^{*}$ equipped with the weak-star topology.

(b) Is the map $𝑥 \mapsto 𝛿_{𝑥}$ continuous if ${(𝐶 ([0, 1]))}^{*}$ equipped with the norm topology? Justify your answer.

Stanford Fall 2025, 3 Real analysis, compact embedding, interpolation inequalities, Banach spaces

(a) Let $(𝑋, {‖ \cdot ‖}_{𝑋})$ , $(𝑌, {‖ \cdot ‖}_{𝑌})$ , $(𝑍, {‖ \cdot ‖}_{𝑍})$ be Banach spaces. Suppose $𝑋 \subset 𝑌$ is compact and $𝑌 \subset 𝑍$ is continuous. Prove that for every $𝜖 > 0$ , there exists $𝐶_{𝜖} > 0$ such that

{‖ 𝑢 ‖}_{𝑌} \leq 𝜖 {‖ 𝑢 ‖}_{𝑋} + 𝐶_{𝜖} {‖ 𝑢 ‖}_{𝑍} \forall 𝑢 \in 𝑋 .

(b) Prove that for every $𝜖 > 0$ , there exists $𝐶_{𝜖} > 0$ such that the following holds for all $𝑓 \in 𝐶^{1} ([0, 1])$ :

max_{[0, 1]} | 𝑓 | \leq 𝜖 max_{[0, 1]} | 𝑓^{'} | + 𝐶_{𝜖} \int_{0}^{1} | 𝑓 | .

UC Irvine Fall 2025, 3 Real analysis, compactness, uniform convergence, continuity

Let $𝐾 \subset ℝ$ be compact, and suppose $𝑓_{𝑛} : 𝐾 \to ℝ$ is a sequence of continuous functions converging uniformly to $𝑓$ . Show that $𝑓$ is continuous.

UC Irvine Fall 2025, 3 Real analysis, local L^1 functions, duality inequalities, compact support

Let $𝑓 \in 𝐿_{1}^{loc} (ℝ^{𝑛})$ , let $𝑝 \in (0, 1)$ , and let $𝐶_{𝑐} (ℝ^{𝑛})$ be the set of continuous functions with compact support in $ℝ^{𝑛}$ . Assume that

\int_{ℝ^{𝑛}} 𝑓 (𝑥) 𝑔 (𝑥) d 𝑥 \leq {(\int_{ℝ^{𝑛}} | 𝑔 (𝑥) |^{𝑝} d 𝑥)}^{1 / 𝑝}

for all $𝑔 \in 𝐶_{𝑐} (ℝ^{𝑛})$ . Prove that

𝑓 (𝑥) = 0

for a.e. $𝑥 \in ℝ^{𝑛}$ .

Stanford Fall 2025, 4 Real analysis, Lebesgue measure, covering arguments, Steinhaus theorem

Prove that for any $𝐴 \subset ℝ$ with positive Lebesgue measure there exist $𝑥 \in 𝐴$ and $𝑦 > 0$ for which $𝑥 + 𝑦, 𝑥 - 𝑦 \in 𝐴$ .

Stanford Fall 2025, 4 Real analysis, smooth functions, Taylor's theorem

Let $𝑓 : [0, 1] \to ℝ$ be a smooth function. Suppose that for every $𝑥 \in [0, 1]$ , there exists $𝑛 \in ℤ_{\geq 0}$ such that $𝑓^{(𝑛)} (𝑥) = 0$ . Prove that $𝑓$ is a polynomial.

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