querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Spring 2024, 1 Real analysis, Schwartz functions, Fourier transform

Suppose that $𝑓$ is a Schwartz function on $ℝ$ with $\int_{- \infty}^{\infty} 𝑥^{𝑘} 𝑓 (𝑥) d 𝑥 = 0$ for all $𝑘 \in {0} \cup ℕ$ . Is $𝑓$ the zero function? Prove this or give a counterexample.

Stanford Spring 2024, 1 Real analysis, weak convergence, bounded operators, strong operator topology

Let $𝐻$ be a Hilbert space.

(a) Suppose ${𝑥_{𝑛}}_{𝑛 = 1}^{\infty} \subset 𝐻$ , $𝑥 \in 𝐻$ . Prove that $𝑥_{𝑛} \to 𝑥$ in norm if and only if $𝑥_{𝑛} \to 𝑥$ weakly and $‖ 𝑥_{𝑛} ‖ \to ‖ 𝑥 ‖$ .

(b) Let ${𝑇_{𝑛}}_{𝑛 = 1}^{\infty} \subset ℒ︀ (𝐻)$ . Prove that $𝑇_{𝑛} \to 𝑇$ in the strong operator topology if and only if $𝑇_{𝑛} \to 𝑇$ in the weak operator topology and $𝑇_{𝑛}^{*} 𝑇_{𝑛} \to 𝑇^{*} 𝑇$ in the weak operator topology.

UC Irvine Spring 2024, 1 Real analysis, sequences, ratio test, root test

Let ${(𝑎_{𝑛})}_{𝑛 = 0}^{\infty}$ be a sequence of positive numbers and suppose

lim_{𝑛 \to \infty} \frac{𝑎_{𝑛 + 1}}{𝑎_{𝑛}} = 𝑎 .

Show that the limit ${lim}_{𝑛 \to \infty} 𝑎_{𝑛}^{1 / 𝑛}$ exists and has the same value $𝑎$ .

UC Irvine Spring 2024, 1 Real analysis, L^2 functions, antiderivatives, growth rate

Assume that $𝑓 \in 𝐿^{2} (ℝ)$ . Let $𝐹 (𝑥) = \int_{0}^{𝑥} 𝑓 (𝑡) d 𝑡$ . Show that

lim_{𝑥 \to \infty} \frac{𝐹 (𝑥)}{\sqrt{𝑥}} = 0 .

Stanford Spring 2024, 2 Real analysis, weak convergence, Lp spaces

Let $1 < 𝑝 < \infty$ . Suppose ${𝑓_{𝑛}}_{𝑛 = 1}^{\infty} \subset 𝐿^{𝑝 ([0, 1])}$ are functions such that for each $𝑛 \in ℕ$ , $𝑓_{𝑛 (𝑥)} \geq 0$ for a.e. $𝑥$ . If $𝑓_{𝑛}$ converges weakly (in $𝐿^{𝑝}$ ) to a function $𝑓 \in 𝐿^{𝑝 ([0, 1])}$ , prove that $𝑓 (𝑥) \geq 0$ for a.e. $𝑥$ .

Stanford Spring 2024, 2 Real analysis, maximal functions, Lebesgue measure

Let $𝑓_{𝑛} : [0, 1] \to [0, 1]$ be a sequence of Lebesgue measurable functions such that ${lim}_{𝑛 \to \infty} 𝑓_{𝑛 (𝑥)} = 0$ almost everywhere. Let

𝑀 𝑓_{𝑛} (𝑥) = sup_{𝑥 \in 𝐼 \subset [0, 1]} \frac{1}{| 𝐼 |} \int_{𝐼} | 𝑓 (𝑦) | d 𝑦,

where the maximum is taken over closed intervals, be the Hardy–Littlewood maximal function. Show that ${lim}_{𝑛 \to \infty} 𝑀 𝑓_{𝑛} = 0$ a.e.

UC Irvine Spring 2024, 2 Real analysis, improper integrals, convergence

For each $𝑞 > 0$ , investigate convergence or divergence of the integral

\int_{1}^{\infty} \frac{𝑥}{1 + 𝑥^{𝑞}} | sin 𝑥 | d 𝑥 .

UC Irvine Spring 2024, 2 Real analysis, Scheffé's lemma, pointwise convergence, L^1 convergence

Suppose that $𝑓 (𝑥)$ and ${𝑓_{𝑛} (𝑥)}_{𝑛 \geq 1}$ are nonnegative integrable functions on $ℝ$ . Assume further that

lim_{𝑛 \to \infty} 𝑓_{𝑛} (𝑥) = 𝑓 (𝑥) a.e. and lim_{𝑛 \to \infty} \int_{ℝ} 𝑓_{𝑛} (𝑥) d 𝑥 = \int_{ℝ} 𝑓 (𝑥) d 𝑥 .

Prove that for any measurable set $𝐸 \subset ℝ$ ,

lim_{𝑛 \to \infty} \int_{𝐸} 𝑓_{𝑛} (𝑥) d 𝑥 = \int_{𝐸} 𝑓 (𝑥) d 𝑥 .

Stanford Spring 2024, 3 Real analysis, locally convex spaces, seminorms, Fréchet spaces, tempered distributions

For $𝑘 \in ℕ$ , define $𝐼_{𝑘} = [- 𝑘, 𝑘] \subset ℝ$ . Endow $𝐶^{\infty (ℝ)}$ with the locally convex space topology given by the seminorms

{‖ 𝑓 ‖}_{𝑘} = \sum_{𝑗 = 0}^{𝑘} sup_{𝑥 \in 𝐼_{𝑘}} | 𝑓^{(𝑗)} (𝑥) |, 𝑘 \in ℕ .

Denote by ${(𝐶^{\infty (ℝ)})}^{*}$ its topological dual. Prove that $Λ \in {(𝐶^{\infty (ℝ)})}^{*}$ if and only if there exists a tempered distribution $𝜆 \in {(𝒮︀ (ℝ))}^{*}$ such that $supp (𝜆) \subset ℝ$ is a bounded set and

𝜆 (𝑓) = Λ (𝑓), \forall 𝑓 \in 𝒮︀ (ℝ) .

(Recall that for a linear map $𝜆 : 𝒮︀ (ℝ) \to ℂ$ , we say $𝑥 \notin supp (𝜆)$ if $\exists$ open set $𝑈_{𝑥}$ , $𝑥 \in 𝑈_{𝑥}$ such that $supp (𝑓) \subset 𝑈_{𝑥} ⟹ 𝜆 (𝑓) = 0$ .)

Stanford Spring 2024, 3 Real analysis, dual spaces, reflexivity, bounded operators

Suppose $𝑋_{1}, 𝑋_{2}, 𝑌$ are reflexive Banach spaces and $𝐴_{𝑗} : 𝑋_{𝑗} \to 𝑌$ are bounded linear maps. Suppose also that there is $𝐶 > 0$ such that

{‖ 𝜆 ‖}_{𝑌^{*}} \leq 𝐶 ({‖ 𝐴_{1}^{*} 𝜆 ‖}_{𝑋_{1}^{*}} + {‖ 𝐴_{2}^{*} 𝜆 ‖}_{𝑋_{2}^{*}}), \forall 𝜆 \in 𝑌^{*},

where $𝑋_{1}^{*}, 𝑋_{2}^{*}, 𝑌^{*}$ denote the dual spaces of $𝑋_{1}, 𝑋_{2}, 𝑌$ , respectively, and $𝐴_{𝑖}^{*}$ is the adjoint of $𝐴_{𝑖}$ .

Show that for all $𝑦 \in 𝑌$ there exist $𝑥_{𝑗} \in 𝑋_{𝑗}$ such that $𝐴_{1} 𝑥_{1} + 𝐴_{2} 𝑥_{2} = 𝑦$ .

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