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Spam yourself with math PhD qualifying-exam questions.

UC Irvine Spring 2025, 6 Real analysis, monotone functions, singular functions, almost-everywhere differentiability

(a) Construct a non-decreasing function $𝑓$ on $[0, 1]$ such that $𝑓^{'} (𝑥) = 0$ for almost every $𝑥 \in [0, 1]$ and

𝑓 (1) - 𝑓 (0) = 1 .

(b) Let $𝐸 \subset [0, 1]$ be a set of Lebesgue measure $0$ . Construct a non-decreasing function $𝑓$ on $[0, 1]$ such that $𝑓^{'} (𝑥)$ does not exist at every point $𝑥 \in 𝐸$ .

Stanford Spring 2026, 1 Real analysis, weak convergence, Lp spaces, Fourier series

Suppose $𝑓_{𝑛}, 𝑓$ are $𝐿^{2}$ functions on the $⊤ = ℝ / ℤ$ , $𝑓_{𝑛} \to 𝑓$ weakly in $𝐿^{2}$ .

(a) Show that the sequence ${| 𝑓_{𝑛} |}^{2}$ is bounded in $𝐿^{1}$ .

(b) Give an example that ${| 𝑓_{𝑛} |}^{2}$ may not converge to ${| 𝑓 |}^{2}$ in $𝐿^{1}$ .

Stanford Spring 2026, 1 Real analysis, Lp spaces, continuity of translation

Let $𝑓 \in 𝐿^{2} (ℝ)$ . Prove that ${lim}_{𝑛 \to \infty} \int_{ℝ} 𝑓 (𝑥) 𝑓 (𝑥 + 𝑛) d 𝑥 = 0$ .

UC Irvine Spring 2026, 1 Real analysis, infinite series, products, convergence

Let ${𝑎_{𝑛}}$ be any sequence of positive real numbers. Denote

𝐵_{0} = 1, 𝐵_{𝑛} = \prod_{𝑘 = 1}^{𝑛} (1 + 𝑎_{𝑘}), 𝑛 \geq 1 .

Prove that the series

\sum_{𝑛 = 1}^{\infty} \frac{𝑎_{𝑛}}{𝐵_{𝑛}}

converges and that its sum is less than or equal to $1$ .

UC Irvine Spring 2026, 1 Real analysis, Lebesgue measure, Chebyshev-type inequalities, L^p spaces

Let $0 \leq 𝑓 \in 𝐿^{3} (ℝ)$ . Prove that

\sum_{𝑛 \in ℕ} 𝑛^{2} 𝑚 ({𝑥 \in ℝ : 𝑓 (𝑥) \geq 𝑛}) < \infty .

Here $𝑚$ denotes Lebesgue measure.

Stanford Spring 2026, 2 Real analysis, Fourier transform, oscillatory integrals

Suppose that $𝑢 (𝑥) = 𝜒 (𝑥) {| 𝑥 |}^{𝜆}$ is a function in $ℝ^{𝑛}$ , where $𝜆 \in ℂ$ and $Re 𝜆 > - 𝑛$ , and where $𝜒 \in 𝐶_{𝑐}^{\infty} (ℝ^{𝑛})$ is a cutoff function which equals $1$ in a neighborhood of $𝑥 = 0$ . Show that there exist a constant $𝐶$ such that the Fourier transform $\hat{𝑢} (𝜉)$ satisfies

| \hat{𝑢} (𝜉) | \leq 𝐶 {| 𝜉 |}^{- 𝑛 - Re 𝜆} for all | 𝜉 | \geq 1 .

Stanford Spring 2026, 2 Real analysis, spectral theory, bounded operators

For $𝜙 \in 𝐿^{\infty} ([0, 1])$ , $𝑋 = 𝐿^{2} ([0, 1])$ , define $𝑀_{𝜙} : 𝑋 \to 𝑋$ to be the operator $𝑢 \mapsto 𝜙 𝑢$ . Construct a $𝜙 \in 𝐿^{\infty} ([0, 1])$ such that $𝑀_{𝜙}$ has the following properties:

Justify your answer. Here, $𝜎$ and $𝜎_{𝑝}$ denote the spectrum and the point spectrum (i.e., set of eigenvalues), respectively, and denotes the closure of $𝜎_{𝑝} (𝑀_{𝜙})$ .

UC Irvine Spring 2026, 2 Real analysis, uniform continuity, limits at infinity

Suppose $𝑓 (𝑥)$ is uniformly continuous on $[0, \infty)$ and satisfies

lim_{𝑛 \to \infty} 𝑓 (𝑥 + 𝑛) = 0

for any $𝑥 \in [0, 1]$ .

(a) Show that

lim_{𝑥 \to + \infty} 𝑓 (𝑥) = 0 .

(b) Does the above claim still hold if uniformly continuous is replaced by continuous? Explain your answer.

UC Irvine Spring 2026, 2 Real analysis, semi-finite measures, measure spaces

Let $(𝑋, 𝑀, 𝜇)$ be a semi-finite measure space, let $𝐸 \in 𝑀$ , and suppose that

𝜇 (𝐸) = \infty .

Show that for any $𝐶 > 0$ , there exists $𝐹 \in 𝑀$ such that

𝐹 \subset 𝐸 and 𝜇 (𝐹) \in (𝐶, \infty) .

Stanford Spring 2026, 3 Real analysis, weak-star topology, dual spaces, density

Let $𝑋$ be a separable Banach space. Show that the unit ball $𝐵^{*}$ in $𝑋^{*}$ is weak-star separable (i.e., there is a countable subset which is dense with respect to the weak-star topology).

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