querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Fall 2009, 3 Real analysis, weak derivatives, absolute continuity

Let $𝑓, 𝑔 \in 𝐿^{1} (⊤)$ , $⊤ = ℝ / (2 𝜋 ℤ)$ , and assume that for any $𝜓 \in 𝐶^{\infty} (⊤)$ ,

\int 𝑓 (𝑡) 𝜓^{'} (𝑡) d 𝑡 = - \int 𝑔 (𝑡) 𝜓 (𝑡) d 𝑡 .

(If this holds, it is common to say that " $𝑔$ is a weak derivative of $𝑓$ ".) Prove that $𝑓$ is absolutely continuous and $𝑓^{'} = 𝑔$ a.e.

Stanford Fall 2009, 3 Real analysis, Volterra operator, spectral radius

For $𝑓 \in 𝐶 ([0, 1])$ , $𝑥 \in [0, 1]$ , let $(𝑇 𝑓) (𝑥)$ be given by

(𝑇 𝑓) (𝑥) = \int_{0}^{𝑥} 𝑓 (𝑦) d 𝑦 .

Show that $𝑇 : 𝐶 ([0, 1]) \to 𝐶 ([0, 1])$ is bounded, find (with proof) $‖ 𝑇 ‖$ and the spectral radius of $𝑇$ .

Stanford Fall 2009, 4 Real analysis, compact operators, adjoint operators

Suppose that $𝑋, 𝑌$ are Banach spaces, $𝑇 \in ℒ︀ (𝑋, 𝑌)$ is compact.

(a) If $𝑆 \in ℒ︀ (𝑋, 𝑋)$ , $𝑅 \in ℒ︀ (𝑌, 𝑌)$ , show that $𝑇 𝑆, 𝑅 𝑇 \in ℒ︀ (𝑋, 𝑌)$ are compact.

(b) Show that $𝑇^{*} \in ℒ︀ (𝑌^{*}, 𝑋^{*})$ is compact.

Stanford Fall 2009, 4 Real analysis, locally convex spaces, seminorms, Fréchet spaces

Suppose that $𝑋$ is a locally convex vector space with topology $𝒯︀$ generated by a family ${𝜌_{𝑎} : 𝑎 \in 𝐴}$ of seminorms.

(a) If $‖ \cdot ‖$ is a continuous seminorm on $𝑋$ , show that there exist $𝑎_{1}, \dots, 𝑎_{𝑛} \in 𝐴$ and $𝐶 > 0$ , such that $‖ 𝑥 ‖ \leq 𝐶 (𝜌_{𝑎_{1}} (𝑥) + \dots + 𝜌_{𝑎_{𝑛}} (𝑥))$ for all $𝑥 \in 𝑋$ .

(b) Let $𝑋 = 𝐶^{\infty (⊤)}$ , $⊤ = ℝ / (2 𝜋 ℤ)$ . Show that $𝑋$ does not have a norm generating its standard topology (given by ${𝜌_{𝑘} = {‖ \cdot ‖}_{𝐶^{𝑘}} : 𝑘 \geq 0, 𝑘 \in ℤ}$ ).

Stanford Fall 2009, 5 Real analysis, uniform convexity, weak convergence, Banach spaces

A Banach space $𝐵$ is uniformly convex if for every $𝜀 \in (0, 1)$ there exists $𝜂 < 1$ such that if $𝑥, 𝑦 \in 𝐵$ , $‖ 𝑥 ‖ = ‖ 𝑦 ‖ = 1$ and $‖ 𝑥 - 𝑦 ‖ > 2 𝜀$ then $‖ \frac{1}{2} (𝑥 + 𝑦) ‖ < 𝜂$ .

Let $𝐵$ be a uniformly convex Banach space.

(a) Assume that $𝑥_{𝑛} \in 𝐵$ for $𝑛 = 1, 2, \dots$ and $𝑥_{𝑛} \to 𝑥_{0}$ in the weak topology, and $‖ 𝑥_{𝑛} ‖ \to ‖ 𝑥_{0} ‖$ . Prove that $𝑥_{𝑛} \to 𝑥_{0}$ in norm.

Hint: Assume that ${𝑥_{𝑛}}$ is not a Cauchy sequence, and for suitable pairs $𝑛_{𝑗}, 𝑚_{𝑗}$ , $𝑛_{𝑗} < 𝑚_{𝑗}$ , $𝑛_{𝑗} \to \infty$ , consider $𝑦_{𝑗} = (𝑥_{𝑛_{𝑗}} + 𝑥_{𝑚_{𝑗}}) / 2$ .

(b) Give an example that the statement in (a) is false for general Banach spaces.

Stanford Fall 2009, 5 Real analysis, Fourier transform, uncertainty principle, Schwartz functions

Let $𝑓 : ℝ \to ℂ$ be a Schwartz function (i.e. $𝑓$ is $𝐶^{\infty}$ and $sup | 𝑥^{𝑁} \frac{d^{𝑘} 𝑓}{d 𝑥^{𝑘}} | < \infty$ for all $𝑁$ and $𝑘$ ) satisfying $\int_{ℝ} {| 𝑓 (𝑥) |}^{2} d 𝑥 = 1$ . Let $\hat{𝑓} (𝑦) = \frac{1}{\sqrt{2 𝜋}} \int_{ℝ} 𝑒^{- 𝑖 𝑥 𝑦} 𝑓 (𝑥) d 𝑥$ denote the Fourier transform of $𝑓$ .

(a) Show that

(\int_{ℝ} {(𝑥 - 𝑥_{0})}^{2} {| 𝑓 (𝑥) |}^{2} d 𝑥) (\int_{ℝ} {(𝑦 - 𝑦_{0})}^{2} {| \hat{𝑓} (𝑦) |}^{2} d 𝑦) \geq \frac{1}{4}

for all $𝑥_{0}, 𝑦_{0} \in ℝ$ .

Hint: You may find it easier to reduce to, and then work with, the case $𝑥_{0} = 𝑦_{0} = 0$ .

(b) Suppose that the equality sign holds in (a). Show that there exists a real number $𝜆 > 0$ and a complex number $𝑐$ such that $| 𝑐 | = 1$ and

𝑓 (𝑥) = 𝑐 {(𝜋 𝜆)}^{- 1 / 4} exp (- \frac{{(𝑥 - 𝑥_{0})}^{2}}{2 𝜆} + 𝑖 𝑥 𝑦_{0})

for all $𝑥 \in ℝ$ .

Stanford Fall 2010, 1 Real analysis, uniform boundedness, bounded operators

Suppose $(𝑋, {‖ \cdot ‖}_{𝑋})$ , $(𝑌, {‖ \cdot ‖}_{𝑌})$ are Banach spaces, and $𝑌$ is a subspace of $𝑋$ with the inclusion $𝜄 : 𝑌 \to 𝑋$ continuous in the respective Banach space topologies. Suppose that $𝑇_{𝑛} \in ℒ︀ (𝑋)$ for $𝑛 \in ℕ$ . Suppose moreover that for each $𝑥 \in 𝑋$ and $𝑛 \in ℕ$ one has $𝑇_{𝑛} 𝑥 \in 𝑌$ , and in addition that for each $𝑥 \in 𝑋$ there exists $𝐶$ (independent of $𝑛$ ) such that ${‖ 𝑇_{𝑛} 𝑥 ‖}_{𝑌} \leq 𝐶$ . Show that for all $𝑛$ , $𝑇_{𝑛} \in ℒ︀ (𝑋, 𝑌)$ , and show that there exists $𝐶$ such that for all $𝑛$ one has ${‖ 𝑇_{𝑛} ‖}_{ℒ︀ (𝑋, 𝑌)} \leq 𝐶$ .

Stanford Fall 2010, 1 Real analysis, Fourier transform, analytic continuation, trigonometric polynomials

Two short problems:

(a) Suppose that $𝑓$ is a compactly supported continuous function on $ℝ^{𝑛}$ (i.e. $𝑓$ vanishes outside a compact set), and suppose that its Fourier transform $\hat{𝑓}$ , given by $\hat{𝑓} (𝜉) = \int_{ℝ^{𝑛}} 𝑒^{- 𝑖 𝑥 \cdot 𝜉} 𝑓 (𝑥) d 𝑥$ , vanishes on a non-empty open set. Show that $𝑓$ is identically $0$ .

(b) Let $⊤ = ℝ / (2 𝜋 ℤ)$ , $1 < 𝑝 < \infty$ . Suppose that $ℎ \in 𝐿^{𝑝 (⊤)}$ , $ℎ$ is non-zero a.e., and let

𝑉 = {𝑃 ℎ : 𝑃 a trigonometric polynomial} \subset 𝐿^{𝑝 (⊤)} .

Show that $𝑉$ is dense in $𝐿^{𝑝 (⊤)}$ .

Stanford Fall 2010, 2 Real analysis, weak sequential compactness, Lp spaces, reflexivity

Consider the spaces $𝐿^{𝑝 ([0, 1])}$ , $1 \leq 𝑝 < \infty$ . For which $𝑝$ is the unit ball, ${𝑓 \in 𝐿^{𝑝} : {‖ 𝑓 ‖}_{𝐿^{𝑝}} \leq 1}$ , weakly sequentially compact, i.e. for which $𝑝$ is it true that if ${𝑓_{𝑛}}_{𝑛 = 1}^{\infty}$ is a sequence in the unit ball in $𝐿^{𝑝}$ then it has a weakly convergent subsequence? For each $𝑝$ , either prove or disprove weak sequential compactness.

Stanford Fall 2010, 2 Real analysis, sequence spaces, density

Let $𝑋$ denote the vector space of all sequences ${𝑎_{𝑛} : 𝑛 \in ℕ}$ with $\sum_{𝑛 = 1}^{\infty} 𝑛 {| 𝑎_{𝑛} |}^{2} < \infty$ .

(a) Prove or disprove: the set $𝑋$ is a dense subset of $ℓ^{2} (ℕ)$ .

(b) Prove or disprove: the set $𝑋$ is a dense subset of $ℓ^{\infty (ℕ)}$ .

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