querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Fall 2013, 3 Real analysis, oscillatory integrals

Suppose $𝑓 \in 𝐶^{\infty (ℝ^{𝑛})}$ is real valued and $𝐾 \subseteq ℝ^{𝑛}$ is compact. Show that if the differential of $𝑓$ does not vanish on $𝐾$ then for all $𝑢 \in 𝐶^{\infty (ℝ^{𝑛})}$ with support in $𝐾$ and for all $𝑁 \geq 0$ there is $𝐶 > 0$ such that

| \int 𝑒^{𝑖 𝜔 𝑓 (𝑥)} 𝑢 (𝑥) d 𝑥 | \leq 𝐶 𝜔^{- 𝑁}, 𝜔 > 1 .

Stanford Fall 2013, 3 Real analysis, Sobolev spaces, compact operators, Schwartz functions

Let $𝒮︀ (ℝ)$ denote the set of Schwartz functions, and ${𝒮︀}^{'} (ℝ)$ the dual space of tempered distributions. For $𝑢 \in {𝒮︀}^{'} (ℝ)$ let $𝐷 𝑢$ denote the distributional derivative of $𝑢$ , and let $𝑀 𝑢$ be the distribution $𝑥 𝑢$ (i.e. $(𝑀 𝑢) (𝜑) = 𝑢 (𝑥 𝜑)$ , $𝜑 \in 𝒮︀ (ℝ)$ ). Let $𝐻 = {𝑢 \in 𝐿^{2} (ℝ) : 𝐷 𝑢 \in 𝐿^{2}, 𝑀 𝑢 \in 𝐿^{2}}$ , equipped with the norm ${‖ 𝑢 ‖}_{𝐻}^{2} = {‖ 𝑢 ‖}_{𝐿^{2}}^{2} + {‖ 𝐷 𝑢 ‖}_{𝐿^{2}}^{2} + {‖ 𝑀 𝑢 ‖}_{𝐿^{2}}^{2}$ . Show that $𝐻$ is a Hilbert space (with the norm being induced by the inner product), $𝒮︀ (ℝ)$ is dense in $𝐻$ , and the inclusion map $𝜄 : 𝐻 \to 𝐿^{2} (ℝ)$ is compact.

Stanford Fall 2013, 4 Real analysis, Schwartz functions, seminorms, Fréchet spaces

Show that the set of Schwartz functions, $𝒮︀ (ℝ)$ , consisting of $𝐶^{\infty}$ functions such that the seminorms , $ℓ, 𝑘 \in ℕ$ , are finite, is not a normed space, i.e. there exists no norm on $𝒮︀ (ℝ)$ giving rise to the locally convex topology induced by these seminorms. (Here $ℕ$ is the set of non-negative integers.)

Stanford Fall 2013, 4 Real analysis, Borel measures, weak-star topology, weak topology

Recall that with $⊤ = ℝ / (2 𝜋 ℤ)$ the circle, the dual space of $𝐶 (⊤)$ is the Banach space $ℳ︀$ of finite Borel measures. Show that $𝐿^{1} (⊤)$ is closed in $ℳ︀$ in the Banach topology induced by this duality, but is not closed in $ℳ︀$ if the latter is equipped with the weak-* topology. Is $𝐿^{1} (⊤)$ closed in $ℳ︀$ if $ℳ︀$ is equipped with the weak topology?

Stanford Fall 2013, 5 Real analysis, Fourier series, commuting operators

Let $⊤ = ℝ / (2 𝜋 ℤ)$ denote the circle. Thus, $𝐶^{𝑘 (⊤)}$ can be identified with the space of $2 𝜋$ -periodic $𝐶^{𝑘}$ functions on $ℝ$ . Let $𝑀 : 𝐶^{𝑘 (⊤)} \to 𝐶^{𝑘 (⊤)}$ denote the multiplication operator by $𝑒^{𝑖 𝑥}$ : $(𝑀 𝑓) (𝑥) = 𝑒^{𝑖 𝑥} 𝑓 (𝑥)$ , and for $𝑘 \geq 1$ (including $𝑘 = \infty$ ) let $\frac{d}{d 𝑥} : 𝐶^{𝑘 (⊤)} \to 𝐶^{𝑘 - 1} (⊤)$ be the usual derivative.

Suppose that $𝑇 : 𝐶^{\infty (⊤)} \to 𝐶^{\infty (⊤)}$ is a linear map (no continuity of any kind is assumed!) and $𝑇$ satisfies $𝑇 \frac{d}{d 𝑥} = \frac{d}{d 𝑥} 𝑇$ and $𝑇 𝑀 = 𝑀 𝑇$ (on $𝐶^{\infty (⊤)}$ ). Show that there exists $𝑐 \in ℂ$ such that $𝑇 = 𝑐 Id$ , i.e. $𝑇 𝑓 = 𝑐 𝑓$ for all $𝑓 \in 𝐶^{\infty (⊤)}$ .

(Hint: First show that if $𝑦 \in ⊤$ and $𝑓 (𝑦) = 0$ then $(𝑇 𝑓) (𝑦) = 0$ ; this shows that $𝑇$ is multiplication by a function.)

Stanford Fall 2013, 5 Real analysis, Borel's lemma

Suppose that ${𝑎_{𝑛} : 𝑛 = 0, 1, 2, \dots}$ is any sequence of real numbers. Show that there exists a real valued function $𝑓 \in 𝐶^{\infty (ℝ)}$ such that $𝑓^{(𝑛)} (0) = 𝑎_{𝑛}$ . (Hint: let $𝜒 \in 𝐶_{𝑐}^{\infty (ℝ)}$ identically 1 near 0, and choose $𝜀_{𝑛} > 0$ appropriately so that ${lim}_{𝑛 \to \infty} 𝜀_{𝑛} = 0$ and

\sum_{𝑛 = 0}^{\infty} \frac{𝑎_{𝑛} 𝑥^{𝑛}}{𝑛!} 𝜒 (𝑥 / 𝜀_{𝑛})

converges in $𝐶^{\infty}$ .)

Stanford Fall 2014, 1 Real analysis, bounded operators, Banach spaces

Prove or disprove: If $𝐻$ is an infinite dimensional separable Hilbert space, then $ℬ︀ (𝐻, 𝐻)$ is separable in the operator-norm topology.

Stanford Fall 2014, 1 Real analysis, Taylor approximation, dual spaces

Two quick problems.

(a) Suppose that $𝑓 : ℝ \to ℝ$ is a function with the property that for all $𝑘 \in ℕ$ and for all $𝑥 \in ℝ$ there exists a polynomial $𝑃_{𝑥, 𝑘}$ on $ℝ$ such that $| 𝑓 (𝑦) - 𝑃_{𝑥, 𝑘} (𝑦) | \leq 𝐶_{𝑥, 𝑘} {| 𝑥 - 𝑦 |}^{𝑘 + 1}$ if $| 𝑥 - 𝑦 | < 1$ . Is $𝑓$ infinitely differentiable? Prove this or give a counterexample.

(b) Suppose that $𝑌$ is a normed complex vector space with norm $‖ \cdot ‖$ , and $𝜆 : 𝑌 \to ℂ$ is linear but is not continuous. Show that $𝑁 = 𝜆^{- 1} ({0})$ is dense in $𝑌$ .

Stanford Fall 2014, 2 Real analysis, spectral theory, compact operators, Lp spaces

Let $(𝑆, 𝜇)$ be a $𝜎$ -finite measure space. Let $𝑓 \in 𝐿^{\infty (𝑆, 𝜇)}$ and let $𝑀_{𝑓}$ be the bounded linear operator on $𝐿^{2} (𝑆, 𝜇)$ , $𝑀_{𝑓} (𝑔) = 𝑓 𝑔$ .

(a) Find a necessary and sufficient condition (in terms of $𝑓$ and $𝜇$ ) for $𝑀_{𝑓}$ to have an eigenvector.

(b) Find a necessary and sufficient condition (in terms of $𝑓$ and $𝜇$ ) for $𝑀_{𝑓}$ to be compact.

Stanford Fall 2014, 2 Real analysis, compact operators, spectral theory

Suppose $𝐴$ is a compact operator on a Hilbert space $𝐻$ . Prove that if $𝑧 𝐼 - 𝐴$ is injective, and $𝑧 \neq 0$ , then the image of $𝑧 𝐼 - 𝐴$ is a closed subspace. Is this also true even if $𝑧 𝐼 - 𝐴$ is not injective?

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