querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Fall 2018, 3 Real analysis, weak topology, locally convex spaces, seminorms

Suppose $𝑋$ is a vector space over $ℂ$ and $ℱ︀$ is a collection of linear maps $𝑋 \to ℂ$ . Equip $𝑋$ with the $ℱ︀$ -weak topology, i.e. the weakest topology in which all elements of $ℱ︀$ are continuous.

(a) Show that the vector space operations $+ : 𝑋 \times 𝑋 \to 𝑋$ and $\cdot : ℂ \times 𝑋 \to 𝑋$ are continuous in this topology (where $𝑋 \times 𝑋$ and $ℂ \times 𝑋$ are equipped with the product topology).

(b) Suppose that $𝜌 : 𝑋 \to [0, \infty)$ is continuous and is a seminorm. Show that there exist $𝑘 \in ℕ$ , $ℓ_{1}, \dots, ℓ_{𝑘} \in ℱ︀$ and $𝐶 > 0$ such that $𝜌 (𝑥) \leq 𝐶 \sum_{𝑗 = 1}^{𝑘} | ℓ_{𝑗} (𝑥) |$ for all $𝑥 \in 𝑋$ .

Stanford Fall 2018, 3 Real analysis, integral operators, Lp spaces, Schur test

Suppose that $𝐾 \in 𝐿^{𝑝 ([0, 1] \times [0, 1])}$ , $1 < 𝑝 < \infty$ . Let $𝑞$ be the dual exponent, $𝑝^{- 1} + 𝑞^{- 1} = 1$ .

(a) For $𝑓 \in 𝐿^{𝑞 ([0, 1])}$ , let $(𝐴 𝑓) (𝑥) = \int 𝐾 (𝑥, 𝑦) 𝑓 (𝑦) d 𝑦$ . Show that $(𝐴 𝑓) (𝑥)$ indeed exists for almost every $𝑥$ and $𝐴 \in ℒ︀ (𝐿^{𝑞 ([0, 1])}, 𝐿^{𝑝 ([0, 1])})$ .

(b) Suppose that for every $𝑓 \in 𝐿^{𝑞 ([0, 1])}$ , $(𝐴 𝑓) (𝑥) = 0$ for almost every $𝑥$ . Show that $𝐾 = 0$ a.e.

Stanford Fall 2018, 4 Real analysis, adjoint operators, bounded operators, Hilbert space geometry

Suppose that $𝑋, 𝑌$ are Hilbert spaces, $𝑃 \in ℒ︀ (𝑋, 𝑌)$ , the set of bounded linear operators $𝑋 \to 𝑌$ . Suppose also that there is $𝐶 \geq 0$ such that ${‖ 𝑥 ‖}_{𝑋} \leq 𝐶 {‖ 𝑃 𝑥 ‖}_{𝑌}$ for all $𝑥 \in 𝑋$ .

(a) Show that $𝑃$ is surjective if and only if $𝑃^{*}$ is injective.

(b) Show that if $𝑃$ is surjective then ${‖ 𝑣 ‖}_{𝑌^{*}} \leq 𝐶 {‖ 𝑃^{*} 𝑣 ‖}_{𝑋^{*}}$ for all $𝑣 \in 𝑌^{*}$ , where $𝐶$ is as above.

Stanford Fall 2018, 4 Real analysis, Fredholm operators, Banach spaces, compact operators

Suppose $𝑇 : 𝑋 \to 𝑌$ is a Fredholm map between two complex Banach spaces, i.e. has closed range, and $Ker 𝑇$ , $𝑌 / Ran 𝑇$ are finite dimensional.

(a) Show that there exist $𝑚, 𝑛 \geq 0$ integers such that $𝑋 \oplus ℂ^{𝑚}$ is isomorphic to $𝑌 \oplus ℂ^{𝑛}$ as Banach spaces (there is an invertible continuous linear map between them).

(b) Prove that $𝑋$ is separable if and only if $𝑌$ is separable.

Stanford Fall 2018, 5 Real analysis, Fourier transform, compact operators, integral operators

Let $𝜒_{[𝛼, 𝛽]}$ denote the characteristic (indicator) function of the interval $[𝛼, 𝛽]$ , as well as the corresponding multiplication operator. Below let $ℱ︀$ be the Fourier transform on $𝐿^{2} (ℝ)$ .

(a) Show that for $𝑎, 𝑏 > 0$ ,

{ℱ︀}^{- 1} 𝜒_{[- 𝑏, 𝑏]} ℱ︀ 𝜒_{[- 𝑎, 𝑎]} \in ℒ︀ (𝐿^{2} (ℝ))

is compact.

(b) Show that for $𝑎, 𝑏 > 0$ ,

{ℱ︀}^{- 1} (1 - 𝜒_{[- 𝑏, 𝑏]}) ℱ︀ 𝜒_{[- 𝑎, 𝑎]} \in ℒ︀ (𝐿^{2} (ℝ))

is not compact.

Stanford Fall 2018, 5 Real analysis, Fourier transform, stationary phase

Let $⊤ = ℝ / (2 𝜋 ℤ)$ be the unit circle, and consider the integral $𝐼 (𝑟) = \int_{⊤} 𝑒^{𝑖 𝑟 cos 𝜃} 𝜑 (𝜃) d 𝜃$ , $𝜑 \in 𝐶^{\infty (⊤)}$ , where $d 𝜃$ is the Lebesgue measure on $⊤$ . Show that there exists $𝐶 > 0$ such that $| 𝐼 (𝑟) | \leq 𝐶 𝑟^{- 1 / 2}$ , $𝑟 \geq 1$ .

Hint: Show that if $𝜑$ is supported away from $[0], [𝜋] \in ℝ / (2 𝜋 ℤ)$ , then $𝐼 (𝑟)$ is rapidly decreasing as $𝑟 \to \infty$ ; then assume $𝜑$ is supported near $[0]$ or $[𝜋]$ , and change variables to obtain an integral of the form $\int 𝑒^{\pm 𝑖 𝑟 𝑠^{2}} \tilde{𝜑} (𝑠) d 𝑠$ (times a prefactor). (Note: $𝐼 (𝑟)$ is essentially the Fourier transform, evaluated at $(𝑟, 0)$ , of a delta distribution on the unit circle in $ℝ^{2}$ .)

Stanford Fall 2019, 1 Real analysis, adjoint operators, Hilbert space geometry, bounded operators

Let $𝐻$ be a Hilbert space. We say that a linear operator $𝐴$ on $𝐻$ is bounded below if there exists $𝑐 > 0$ such that for all $𝑥 \in 𝐻$ , $𝑐 ‖ 𝑥 ‖ \leq ‖ 𝐴 𝑥 ‖$ .

Suppose $𝑇 : 𝐻 \to 𝐻$ is a bounded linear operator. Prove that $𝑇$ is invertible if and only if $𝑇$ and $𝑇^{*}$ are both bounded below.

Stanford Fall 2019, 1 Real analysis, Stone-Weierstrass, compact operators, density

Two short problems:

(a) For a topological space $𝑀$ , let $𝐶 (𝑀)$ denote the vector space of real valued continuous functions on $𝑀$ .

Suppose $𝑋, 𝑌$ are compact Hausdorff topological spaces. Let $𝐷$ be the linear span of functions of the form $𝑢 (𝑥, 𝑦) = 𝜑 (𝑥) 𝜓 (𝑦)$ , $𝜑 \in 𝐶 (𝑋)$ , $𝜓 \in 𝐶 (𝑌)$ . Show that $𝐷$ is dense in $𝐶 (𝑋 \times 𝑌)$ .

(b) Let $𝑋$ be a separable Hilbert space. Show that if $𝐾$ is a compact operator on $𝑋$ , then $𝐾$ is the norm limit of finite rank operators.

Stanford Fall 2019, 2 Real analysis, Lp spaces, Lebesgue measure, norm equivalence

Let $𝑋 \subseteq ℝ$ be a Borel set and $𝜇$ be the Lebesgue measure. Suppose there exist $1 \leq 𝑝 < 𝑞 < + \infty$ such that $𝐿^{𝑞 (𝑋, 𝜇)} \subseteq 𝐿^{𝑝 (𝑋, 𝜇)}$ . Prove that $𝜇 (𝑋) < + \infty$ .

Hint: First show that the inclusion is continuous.

Stanford Fall 2019, 2 Real analysis, locally convex spaces, seminorms

Let $𝑋$ be a complex vector space. Suppose that ${𝜌_{𝛼} : 𝛼 \in 𝐴}$ is a collection of seminorms on $𝑋$ such that for each $𝑥 \in 𝑋 - {0}$ there is $𝛼 \in 𝐴$ such that $𝜌_{𝛼 (𝑥)} \neq 0$ , and $𝐵 : 𝑋 \times 𝑋 \to ℂ$ is a (jointly) continuous bilinear map in the locally convex topology generated by the $𝜌_{𝛼}$ . Show that there exist $𝛼_{1}, \dots, 𝛼_{𝑛} \in 𝐴$ , $𝐶 > 0$ , such that for all $𝑥, 𝑦 \in 𝑋$ ,

| 𝐵 (𝑥, 𝑦) | \leq 𝐶 (𝜌_{𝛼_{1}} (𝑥) + \dots + 𝜌_{𝛼_{𝑛}} (𝑥)) (𝜌_{𝛼_{1}} (𝑦) + \dots + 𝜌_{𝛼_{𝑛}} (𝑦)) .

Questions per page