querent

Consider the operator $𝑃=-\frac{d}{d𝑥}𝑝\frac{d}{d𝑥}+𝑞$ on $\top =ℝ/(2𝜋\mathbb{Z})$, $𝑝,𝑞\in {𝐶}^0(\top )$ are real valued (acting as multiplication operators), and where $𝑝>0$. Let be the ${𝐿}^2(\top )$ inner product.

(a) Show that $𝑃:{𝐻}^1(\top )\to {𝐻}^{-1}(\top )$ is continuous, and holds for $𝑢,𝑣\in {𝐻}^1(\top )$.

(b) Show that there exist $𝐶>0$ and ${𝐶}^{\prime }\ge 0$ such that for $𝑢\in {𝐻}^1(\top )$, , and if $𝑞>0$, one can take ${𝐶}^{\prime }=0$.

(c) Show that ${\Vert 𝑢\Vert }_{{𝐻}^1}\le {𝐶}_1({\Vert 𝑃𝑢\Vert }_{{𝐻}^{-1}}+{\Vert 𝑢\Vert }_{{𝐿}^2})$, and if ${𝐶}^{\prime }=0$ then in fact ${\Vert 𝑢\Vert }_{{𝐻}^1}\le {𝐶}_1{\Vert 𝑃𝑢\Vert }_{{𝐻}^{-1}}$.

(d) Show that if ${𝐶}^{\prime }=0$ then $𝑃:{𝐻}^1\to {𝐻}^{-1}$ is invertible.

(e) Show that even if ${𝐶}^{\prime }\ne 0$, $Ker𝑃\subseteq {𝐻}^1$ is finite dimensional, $Ran𝑃\subseteq {𝐻}^{-1}$ is closed, and has finite codimension.

Prove that if $𝐻$ is a nonseparable Hilbert space and $𝐴$ is a compact operator on $𝐻$, then the nullspace $Ker(𝐴)$ is itself a nonseparable Hilbert space.

Let ${\{({𝑋}_{𝑗},{𝑑}_{𝑗})\}}_{𝑗=1,2}$ be metric spaces and suppose there is a continuous surjection $𝑓:({𝑋}_1,{𝑑}_1)\to ({𝑋}_2,{𝑑}_2)$ satisfying ${𝑑}_2(𝑓(𝑦),𝑓(𝑥))\ge {𝑑}_1(𝑦,𝑥)$ for all $𝑥,𝑦\in {𝑋}_1$. Prove or give a counterexample:

(a) If $({𝑋}_1,{𝑑}_1)$ is complete, then so is $({𝑋}_2,{𝑑}_2)$.

(b) If $({𝑋}_2,{𝑑}_2)$ is complete, then so is $({𝑋}_1,{𝑑}_1)$.

Let $𝐴$ be an infinite closed subset of ${ℝ}^{𝑛}$. Show that there exists a countable set whose closure is $𝐴$.

Compute $sup{\int }_0^1{𝑥}^3𝑔(𝑥)d𝑥$, where the supremum is taken over $𝑔\in {𝐿}^2([0,1])$ with

Justify your answer.

Let $𝑓\in {𝐿}^1([0,1])$ and let $1<𝑝<\infty$. Prove that $𝑓\in {𝐿}^{𝑝([0,1])}$ if and only if

where the supremum is taken over all finite partitions of $[0,1]$ into intervals $\{{𝐼}_{𝑗}\}$.

Consider the sequence recursively defined by

with

Show that ${\{{𝑥}_{𝑛}\}}_{𝑛\in \mathbb{N}}$ converges and determine its limit.

Let $𝜇$ be a Borel measure on ${ℝ}^{𝑛}$. Suppose that $𝜇$ is singular with respect to the Lebesgue measure ${𝑚}_{𝑛}$ and define

Prove that $𝜇\{𝑥:𝑀𝜇(𝑥)<\infty \}=0$.

Suppose that $𝐻$ is a Hilbert space, $𝐴\in \mathrm{ℒ︀}(𝐻)$ is self-adjoint, and for $𝑡\in ℝ$ define $𝑈(𝑡)={𝑓}_{𝑡(𝐴)}$ via the functional calculus where ${𝑓}_{𝑡(𝑠)}={𝑒}^{𝑖𝑡𝑠}$.

(a) Show that for $𝑡\in ℝ$, ${𝑥}_0\in 𝐻$, $𝑥(𝑡)=𝑈(𝑡){𝑥}_0$ satisfies $𝑥\in {𝐶}^1(ℝ;𝐻)$, $𝑥(0)={𝑥}_0$, $\frac{d𝑥}{d𝑡}=𝑖𝐴𝑥$.

(b) Show also that $𝑈(𝑡)$ is unitary for $𝑡\in ℝ$, and $𝑈(𝑡)-𝐼$ is compact for all $𝑡\in ℝ$ if and only if $𝐴$ is compact.

Show that the sum

converges uniformly on $[-2,2]$ to a continuous function denoted by $𝑓$. Give a formula for ${𝑓}^{\prime }(𝑥)$ and prove that your formula is correct.