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Spam yourself with math PhD qualifying-exam questions.

Stanford Fall 2021, 3 Real analysis, Sobolev spaces, distributional derivatives, Lp spaces

Recall that $𝑊^{1, 𝑝} = 𝑊^{1, 𝑝} (ℝ)$ is the collection of $𝐿^{𝑝 (ℝ)}$ functions $𝑓$ such that the distributional derivative $𝑓^{'}$ of $𝑓$ is also in $𝐿^{𝑝}$ . If $𝑢 \in 𝑊^{1, 𝑝}$ show that $| 𝑢 | \in 𝑊^{1, 𝑝}$ and compute its distributional derivative.

Stanford Fall 2021, 3 Real analysis, Banach spaces, bounded operators

Suppose $𝑋$ is a Banach space, $𝐴 \in ℒ︀ (𝑋)$ and $‖ 𝐴 ‖ < 1$ . Show that there exists $𝐵 \in ℒ︀ (𝑋)$ such that $𝐵^{2} = 𝐼 + 𝐴$ .

Stanford Fall 2021, 4 Real analysis, distributions, weak-star topology, Fourier series

Let ${𝒟︀}^{'} (⊤)$ denote the set of distributions on $⊤ = ℝ / (2 𝜋 ℤ)$ , i.e. the dual of $𝐶^{\infty (⊤)}$ , equipped with the "weak-* topology", i.e. the weakest topology in which the maps $𝐸_{𝜑} : {𝒟︀}^{'} (⊤) \to ℂ$ , $𝐸_{𝜑 (𝑢)} = 𝑢 (𝜑)$ are continuous for all $𝜑 \in 𝐶^{\infty (⊤)}$ .

(a) For $𝜑 \in 𝐶^{\infty (⊤^{2})}$ let $𝜑_{𝑥 (𝑦)} = 𝜑 (𝑦, 𝑥)$ , $(𝑦, 𝑥) \in ⊤ \times ⊤$ , so $𝜑_{𝑥} \in 𝐶^{\infty (⊤)}$ . Show that if $𝑢 \in {𝒟︀}^{'} (⊤)$ and $𝜑 \in 𝐶^{\infty (⊤^{2})}$ then the function $𝑓 : ⊤ \to ℂ$ defined by $𝑓 (𝑥) = 𝑢 (𝜑_{𝑥})$ is $𝐶^{\infty}$ .

(b) Show that the convolution map $* : 𝐶 (⊤) \times 𝐶 (⊤) \to 𝐶 (⊤)$ given by

* (𝑓, 𝑔) (𝑥) = \frac{1}{2 𝜋} \int_{0}^{2 𝜋} 𝑓 (𝑥 - 𝑦) 𝑔 (𝑦) d 𝑦

(usually written as $(𝑓 * 𝑔) (𝑥)$ ) has a unique separately continuous extension to a map $\tilde{*} : {𝒟︀}^{'} (⊤) \times {𝒟︀}^{'} (⊤) \to {𝒟︀}^{'} (⊤)$ .

Stanford Fall 2021, 4 Real analysis, norm equivalence, compact operators

Let $𝐶^{𝛼 ([0, 1])}$ , $0 < 𝛼 < 1$ , denote the space of Hölder continuous functions on $[0, 1]$ , i.e. continuous functions such that ${‖ 𝑓 ‖}_{𝛼} = sup | 𝑓 | + {sup}_{𝑥 \neq 𝑦} \frac{| 𝑓 (𝑥) - 𝑓 (𝑦) |}{{| 𝑥 - 𝑦 |}^{𝛼}} < \infty$ , equipped with the norm ${‖ \cdot ‖}_{𝛼}$ .

Suppose that $0 < 𝛽 < 𝛼 < 1$ , ${‖ 𝑓_{𝑛} ‖}_{𝐶^{𝛼 ([0, 1])}} \leq 1$ , $𝑛 \in ℕ$ . Show that there is a subsequence ${𝑓_{𝑛_{𝑖}}}_{𝑖 = 1}^{\infty}$ that converges in $𝐶^{𝛽 ([0, 1])}$ .

Stanford Fall 2021, 5 Real analysis, adjoint operators, compact operators, Fredholm operators

Suppose that $𝐻_{1}$ and $𝐻_{2}$ are separable Hilbert spaces and $𝐴 : 𝐻_{1} \to 𝐻_{2}$ is a bounded linear operator. Suppose that there exist $𝐵 \in ℒ︀ (𝐻_{2}, 𝐻_{1})$ and a compact operator $𝐸$ on $𝐻_{1}$ , such that $𝐵 𝐴 = 𝐼 - 𝐸$ , where $𝐼$ is the identity operator on $𝐻_{1}$ .

(a) Show that the nullspace of $𝐴$ is finite dimensional and the range of $𝐴$ is closed in $𝐻_{2}$ .

(b) Give an example of such operators $𝐴, 𝐵, 𝐸$ (and spaces $𝐻_{1}, 𝐻_{2}$ ) such that the orthocomplement of the range of $𝐴$ is infinite dimensional.

Stanford Fall 2021, 5 Real analysis, elliptic operators, Sobolev spaces, tempered distributions

Suppose that $𝑃 (𝜉) = \sum_{| 𝛼 | \leq 𝑚} 𝑎_{𝛼} 𝜉^{𝛼}$ , $𝑎_{𝛼} \in ℂ$ , is a polynomial of degree $𝑚$ on $ℝ^{𝑛}$ ; here for $𝛼 \in ℕ^{𝑛}$ , $| 𝛼 | = \sum_{𝑗 = 1}^{𝑛} 𝛼_{𝑗}$ , and $𝜉^{𝛼} = 𝜉_{1}^{𝛼_{1}} \dots 𝜉_{𝑛}^{𝛼_{𝑛}}$ . Let $𝑃 (𝐷)$ be the corresponding differential operator, $𝑃 (𝐷) = \sum_{| 𝛼 | \leq 𝑚} 𝑎_{𝛼} 𝐷^{𝛼}$ , , $𝐷^{𝛼} = 𝐷_{1}^{𝛼_{1}} \dots 𝐷_{𝑛}^{𝛼_{𝑛}}$ . We say that $𝑃$ is elliptic if $ℝ^{𝑛} ∋ 𝜉 \neq 0$ implies $\sum_{| 𝛼 | = 𝑚} 𝑎_{𝛼} 𝜉^{𝛼} \neq 0$ .

(a) Show that if $𝑃$ is elliptic, $𝑢 \in {𝒮︀}^{'} (ℝ^{𝑛})$ and $𝑃 𝑢 \in 𝒮︀ (ℝ^{𝑛})$ then $𝑢 \in 𝐶^{\infty (ℝ^{𝑛})}$ .

(b) Recall that for $𝑚 \geq 0$ , $𝐻^{𝑚 (⊤^{𝑛})}$ is the subset of $𝐿^{2} (⊤^{𝑛})$ consisting of functions whose Fourier coefficients satisfy $\sum_{𝑘 \in ℤ^{𝑛}} {(1 + {| 𝑘 |}^{2})}^{𝑚} {| \hat{𝑓} (𝑘) |}^{2} < \infty$ . Here $⊤ = ℝ / (2 𝜋 ℤ)$ and $\hat{𝑓} (𝑘) = {(2 𝜋)}^{- 𝑛 / 2} \int 𝑒^{- 𝑖 𝑥 \cdot 𝑘} 𝑓 (𝑥) d 𝑥$ , $𝑘 \in ℤ^{𝑛}$ .

Show that if $𝑃$ is elliptic of order $𝑚$ then $𝑃$ considered as an operator $𝑃 : 𝐻^{𝑚 (⊤^{𝑛})} \to 𝐿^{2} (⊤^{𝑛})$ has finite dimensional nullspace.

Stanford Fall 2022, 1 Real analysis, self-adjoint operators, spectral theory, Hilbert space geometry

Let $𝑌 \subseteq ℝ$ be a non-empty compact subset. Show that there is a Hilbert space $𝐻$ and a self-adjoint bounded linear operator $𝑇 : 𝐻 \to 𝐻$ such that the spectrum of $𝑇$ is exactly $𝑌$ .

Stanford Fall 2022, 1 Real analysis, Fourier series, trigonometric polynomials

If $𝐴 \subseteq [0, 2 𝜋]$ is measurable, prove that ${lim}_{𝑛 \to \infty} \int_{𝐴} cos 𝑛 𝑥 d 𝑥 = 0$ .

UC Irvine Fall 2022, 1 Real analysis, axiom of completeness, monotone sequences

(a) State the Axiom of Completeness for $ℝ$ .

(b) Prove that a monotone increasing sequence of real numbers converges if and only if it is bounded above.

Stanford Fall 2022, 2 Real analysis, Fréchet spaces, seminorms, norm equivalence

Consider $𝐶^{\infty (⊤)}$ (where $⊤ = ℝ / ℤ$ ) with the usual Fréchet topology given by the seminorms $𝜌_{𝑘 (𝑓)} = {sup}_{𝑥 \in ⊤} | 𝑓^{(𝑘)} (𝑥) |$ . Prove that there is no norm on $𝐶^{\infty (⊤)}$ which induces the same topology.

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