querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Fall 2024, 3 Real analysis, Fubini's theorem, change of variables

Let $𝑓 : [0, 1] \to [0, \infty)$ be a measurable function. Show that the following statements are equivalent:

(a) $\int_{[0, 1] \times [0, 1]} 𝑓 (𝑥^{2} 𝑦) d 𝑥 d 𝑦 < + \infty$ .

(b) $\int_{[0, 1] \times [1 / 2, 1]} 𝑓 (𝑥^{2} 𝑦) d 𝑥 d 𝑦 < + \infty$ .

UC Irvine Fall 2024, 3 Real analysis, compactness, sequence spaces, sumsets

Let $𝐾_{1}, 𝐾_{2} \subset ℓ^{2} (ℕ)$ be compact. Define

𝑆 = {𝑣 \in ℓ^{2} (ℕ) ∣ 𝑣 = 𝑣_{1} + 𝑣_{2} for some 𝑣_{1} \in 𝐾_{1}, 𝑣_{2} \in 𝐾_{2}} .

Prove that $𝑆$ is compact.

UC Irvine Fall 2024, 3 Real analysis, layer-cake formula, distribution functions, L^p bounds

Fix $𝑝 > 1$ , and suppose that $𝑢 \in 𝐿^{𝑝} ([0, 1])$ is nonnegative and that

\int_{{𝑢 \geq 𝑡}} 𝑢 d 𝑥 \leq min {1, 𝑡^{- 𝑝}}

for all $𝑡 > 0$ . Prove that

\int_{0}^{1} 𝑢^{𝑝} d 𝑥 \leq 𝑝 .

Stanford Fall 2024, 4 Real analysis, tempered distributions, distributions

(a) Define the linear map $𝑆_{1} : 𝐶_{𝑐}^{\infty} (ℝ; ℝ) \to ℝ$ by

𝑆_{1} (𝑓) = \int_{- \infty}^{\infty} 𝑓 (𝑥) 𝑒^{𝑥} d 𝑥 \forall 𝑓 \in 𝐶_{𝑐}^{\infty} (ℝ; ℝ) .

Prove that there exists no tempered distribution $𝑇_{1} \in {𝒮︀}^{'} (ℝ; ℝ)$ such that $𝑇_{1} (𝑓) = 𝑆_{1} (𝑓)$ for all $𝑓 \in 𝐶_{𝑐}^{\infty} (ℝ; ℝ)$ .

(b) Define the linear map $𝑆_{2} : 𝐶_{𝑐}^{\infty} (ℝ; ℝ) \to ℝ$ by

𝑆_{2} (𝑓) = \int_{- \infty}^{\infty} 𝑓 (𝑥) 𝑒^{𝑥} cos (𝑒^{𝑥}) d 𝑥, \forall 𝑓 \in 𝐶_{𝑐}^{\infty} (ℝ; ℝ) .

Prove that there exists a tempered distribution $𝑇_{2} \in {𝒮︀}^{'} (ℝ; ℝ)$ such that $𝑇_{2} (𝑓) = 𝑆_{2} (𝑓)$ for all $𝑓 \in 𝐶_{𝑐}^{\infty} (ℝ; ℝ)$ .

Stanford Fall 2024, 4 Real analysis, Sobolev spaces, trace theorem, weak sequential compactness

Let ${𝑓_{𝑛 (𝑥, 𝑦, 𝑧)}}_{𝑛 = 1}^{\infty}$ be a sequence of continuously differentiable functions on $ℝ^{3}$ which satisfy

{(\int_{ℝ^{3}} ({| \nabla 𝑓_{𝑛} |}^{2} + {| 𝑓_{𝑛} |}^{2}) (𝑥, 𝑦, 𝑧) d 𝑥 d 𝑦 d 𝑧)}^{1 / 2} \leq 1 .

(a) Prove that ${𝑓_{𝑛 (𝑥, 𝑦, 0)}}_{𝑛 = 0}^{\infty}$ is a bounded sequence on the ${𝑧 = 0}$ plane with respect to the $𝐿^{2} (d 𝑥 d 𝑦)$ norm.

(b) Suppose that

{(\int_{ℝ^{3}} ({| \nabla 𝑓_{𝑛} |}^{2} + (1 + 𝑥^{2} + 𝑦^{2} + 𝑧^{2}) {| 𝑓_{𝑛} |}^{2}) (𝑥, 𝑦, 𝑧) d 𝑥 d 𝑦 d 𝑧)}^{1 / 2} \leq 10 .

Prove that there is a subsequence ${𝑓_{𝑛_{𝑘}} (𝑥, 𝑦, 0)}_{𝑘 = 0}^{\infty}$ which is convergent on the ${𝑧 = 0}$ plane with respect to the $𝐿^{2} (d 𝑥 d 𝑦)$ norm.

UC Irvine Fall 2024, 4 Real analysis, concavity, monotonicity, uniform continuity

Let $𝑓 : (0, \infty) \to (0, \infty)$ be twice differentiable with $𝑓^{″} < 0$ .

(a) Prove that $𝑓$ is increasing.

(b) Prove that $𝑓$ is uniformly continuous.

UC Irvine Fall 2024, 4 Real analysis, weak-* convergence, oscillating functions

Let

𝑔 (𝑥) = {\begin{cases} 1 & if 𝑥 \in [𝑘, 𝑘 + 2 / 3] \\ 𝑘 \in ℤ \\ - 1 & otherwise \end{cases}

and let $𝑔_{𝑗} (𝑥) = 𝑔 (𝑗 𝑥)$ , $𝑗 \in ℕ$ . Prove that for all $𝑓 \in 𝐿^{1} ([0, 1])$ ,

lim_{𝑗 \to \infty} \int_{0}^{1} 𝑔_{𝑗} (𝑥) 𝑓 (𝑥) d 𝑥 = \frac{1}{3} \int_{0}^{1} 𝑓 (𝑥) d 𝑥 .

Stanford Fall 2024, 5 Real analysis, cluster points, countable sets

Suppose $𝑓 : [0, 1] \to [0, \infty]$ has the property that if $𝑓 (𝑦) \neq \infty$ , then ${lim inf}_{𝑥 \to 𝑦} 𝑓 (𝑥) > 𝑓 (𝑦)$ . Prove that ${𝑦 \in [0, 1] : 𝑓 (𝑦) < \infty}$ is (at most) countable.

Stanford Fall 2024, 5 Real analysis, bounded operators, Fredholm operators

Suppose $𝑇 : 𝑋 \to 𝑌$ is a bounded linear operator between Hilbert spaces and $𝑇^{*} 𝑇$ is Fredholm (i.e., it has closed range, as well as finite-dimensional kernel and co-kernel).

(a) Show that $𝑇$ has closed range.

(b) Is $𝑇$ necessarily Fredholm? Prove it or give a counterexample.

UC Irvine Fall 2024, 5 Real analysis, contraction mappings, fixed points, true or false

Which of the following statements are true? Explain your answer.

(a) If $(𝑀, 𝑑)$ is a metric space, and $𝑓 : 𝑀 \to 𝑀$ is a map such that for any distinct $𝑥, 𝑦 \in 𝑀$ one has

𝑑 (𝑓 (𝑥), 𝑓 (𝑦)) < 𝑑 (𝑥, 𝑦),

then $𝑓$ has exactly one fixed point in $𝑀$ .

(b) If $(𝑀, 𝑑)$ is a complete metric space, and $𝑓 : 𝑀 \to 𝑀$ is a map such that

𝑑 (𝑓 (𝑥), 𝑓 (𝑦)) < 𝑑 (𝑥, 𝑦)

for any distinct $𝑥, 𝑦 \in 𝑀$ , then $𝑓$ has exactly one fixed point in $𝑀$ .

(c) If $(𝑀, 𝑑)$ is a bounded metric space, and $𝑓 : 𝑀 \to 𝑀$ is a map such that

𝑑 (𝑓 (𝑥), 𝑓 (𝑦)) < 𝑑 (𝑥, 𝑦)

for any distinct $𝑥, 𝑦 \in 𝑀$ , then $𝑓$ has exactly one fixed point in $𝑀$ .

(d) If $(𝑀, 𝑑)$ is a compact metric space, and $𝑓 : 𝑀 \to 𝑀$ is a map such that

𝑑 (𝑓 (𝑥), 𝑓 (𝑦)) < 𝑑 (𝑥, 𝑦)

for any distinct $𝑥, 𝑦 \in 𝑀$ , then $𝑓$ has exactly one fixed point in $𝑀$ .

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