querent

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

(a) Show that every Hilbert space is uniformly convex.

(b) Let $𝑋$ be a uniformly convex Banach space. Suppose that $𝐶$ is a closed convex subset of $𝑋$, and $𝑧\in 𝑋$. Show that there exists a unique $𝑥\in 𝐶$ such that $\Vert 𝑥-𝑧\Vert =inf\{\Vert 𝑦-𝑧\Vert :𝑦\in 𝐶\}$.

(c) Give an example showing that the property in (b) can fail if $𝑋$ is not uniformly convex.

Consider ${\ell }^2$ with norm ${\Vert {\{{𝑎}_{𝑛}\}}_{𝑛=1}^{\infty }\Vert }^2={\sum }_{𝑛=1}^{\infty }{|{𝑎}_{𝑛}|}^2$. Let $𝑇$ be the operator on ${\ell }^2$ given by $𝑇(\{{𝑎}_1,{𝑎}_2,\dots \})=\left\{{𝑎}_2,\frac{1}{2}{𝑎}_3,\frac{1}{3}{𝑎}_4,\dots \right\}$, i.e. $𝑇({\{{𝑎}_{𝑛}\}}_{𝑛=1}^{\infty })={\{{𝑛}^{-1}{𝑎}_{𝑛+1}\}}_{𝑛=1}^{\infty }$. Show that $𝑇$ is compact, and find the spectrum of $𝑇$. Does $𝑇$ have any eigenvalues?

A projection $𝑃\in \mathrm{ℒ︀}(𝑋)$ on a Hilbert space $𝑋$ is an operator with ${𝑃}^2=𝑃$. Show that such a projection $𝑃$ is an orthogonal projection (i.e. $𝑥-𝑃(𝑥)\in {(𝑃(𝑋))}^{⟂}$ for all $𝑥\in 𝑋$) if and only if $𝑃$ is self-adjoint.

(a) Show that the set of points $𝐴$ at which a function $𝑓:[0,1]\to ℝ$ is continuous is a ${𝐺}_{𝛿}$ set (countable intersection of open sets).

(b) Give an example of a function $𝑓:[0,1]\to ℝ$ that is continuous at all the irrationals and is discontinuous at all the rationals.

(c) Show that there is no function $𝑓:[0,1]\to ℝ$ which is continuous at all the rationals and is discontinuous at all the irrationals.

Let $𝑋$ be a vector space over $ℂ$, $\mathrm{ℱ︀}$ a vector space of linear maps $𝑋\to ℂ$, and equip $𝑋$ with the weakest topology in which all members of $\mathrm{ℱ︀}$ are continuous. Show that the only continuous linear maps $𝑋\to ℂ$ are those in $\mathrm{ℱ︀}$.

Let ${\top }^{𝑛}={(ℝ/(2𝜋\mathbb{Z}))}^{𝑛}$ be the $𝑛$-torus, ${𝑒}_{𝑗}$ the image of the $𝑗$th unit vector in ${ℝ}^{𝑛}$ in the torus, and let ${\mathrm{𝒟︀}}^{\prime }({\top }^{𝑛})$ denote the set of distributions on the torus, equipped with the weak-* topology.

(a) Show that for all $𝑦\in {\top }^{𝑛}$ the map ${𝐿}_{𝑦}:𝐶({\top }^{𝑛})\to 𝐶({\top }^{𝑛})$ defined by $({𝐿}_{𝑦}𝑓)(𝑥)=𝑓(𝑥+𝑦)$ has a unique continuous extension to a map ${\mathrm{𝒟︀}}^{\prime }({\top }^{𝑛})\to {\mathrm{𝒟︀}}^{\prime }({\top }^{𝑛})$.

(b) If $𝑢\in {\mathrm{𝒟︀}}^{\prime }({\top }^{𝑛})$ (i.e. $𝑢$ is a distribution on the torus) then for all $𝑗$, ${𝑢}_{ℎ}={ℎ}^{-1}({𝐿}_{ℎ{𝑒}_{𝑗}}𝑢-𝑢)$ converges to the distributional derivative as $ℎ\to 0$.

(c) For $𝜑\in {𝐶}^{\infty ({\top }^{𝑛+𝑚})}$ let ${𝜑}_{𝑥}(𝑦)=𝜑(𝑦,𝑥)$, $(𝑦,𝑥)\in {\top }^{𝑛}\times {\top }^{𝑚}$, so ${𝜑}_{𝑥}\in {𝐶}^{\infty ({\top }^{𝑛})}$. Show that if $𝑢\in {\mathrm{𝒟︀}}^{\prime }({\top }^{𝑛})$ and $𝜑\in {𝐶}^{\infty ({\top }^{𝑛+𝑚})}$ then the function $𝑓:{\top }^{𝑚}\to ℂ$ defined by $𝑓(𝑥)=𝑢({𝜑}_{𝑥})$ is ${𝐶}^{\infty }$.

Suppose $𝑋,𝑌$ are (reflexive) Banach spaces, ${𝐴}_{𝑛}\in \mathrm{ℒ︀}(𝑋,𝑌)$ for $𝑛\in \mathbb{N}$. Suppose that for all $𝑥\in 𝑋$ and for all $\ell \in {𝑌}^{\ast }$, ${lim}_{𝑛\to \infty }\ell ({𝐴}_{𝑛}𝑥)$ exists. Show that there exists $𝐴\in \mathrm{ℒ︀}(𝑋,𝑌)$ such that ${𝐴}_{𝑛}\to 𝐴$ in the weak operator topology. Give an example of ${𝐴}_{𝑛}$ satisfying these assumptions such that ${𝐴}_{𝑛}$ does not converge to $𝐴$ in the strong operator topology.

For $𝐸\subseteq ℝ$ let $𝐸+𝐸=\{𝑥+𝑦:𝑥,𝑦\in 𝐸\}$, and define $𝐸-𝐸$ similarly. Show that if $𝐸$ is a measurable subset of $ℝ$ of positive Lebesgue measure then $𝐸-𝐸$ and $𝐸+𝐸$ contain non-empty open sets.

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

(a) Suppose $0<𝛼<1$, $𝑋$ is a subspace of ${𝐶}^{𝛼([0,1])}$, and $𝑋$ is closed as a subspace of $𝐶([0,1])$. Show that $𝑋$ is finite dimensional.

(b) Suppose that $𝑋$ is a subspace of $𝐶([0,1])$ and $𝑋$ is closed as a subspace of ${𝐿}^2([0,1])$ (with the standard norm). Show that $𝑋$ is finite dimensional.

Let $\top =ℝ/(2𝜋\mathbb{Z})$, and for $𝑓\in {𝐿}^2({\top }^{𝑛})$ let $\widehat{𝑓}(𝑘)={(2𝜋)}^{-𝑛/2}{\int }_{{\top }^{𝑛}}{𝑒}^{-𝑖𝑥\cdot 𝑘}𝑓(𝑥)d𝑥$, $𝑘\in {\mathbb{Z}}^{𝑛}$, denote the Fourier coefficients of $𝑓$. For $𝑚\ge 0$, ${𝐻}^{𝑚({\top }^{𝑛})}$ is the subset of ${𝐿}^2({\top }^{𝑛})$ consisting of $𝑓\in {𝐿}^2({\top }^{𝑛})$ with Fourier coefficients satisfying ${\sum }_{𝑘\in {\mathbb{Z}}^{𝑛}}{(1+{|𝑘|}^2)}^{𝑚}{\left|\widehat{𝑓}(𝑘)\right|}^2<\infty$, and with norm ${\Vert 𝑓\Vert }_{{𝐻}^{𝑚}({\top }^{𝑛})}^2={\sum }_{𝑘\in {\mathbb{Z}}^{𝑛}}{(1+{|𝑘|}^2)}^{𝑚}{\left|\widehat{𝑓}(𝑘)\right|}^2$.

Suppose that $𝑠>𝑟\ge 0$, $𝑚\ge 0$, and $𝑃:{𝐻}^{𝑠({\top }^{𝑛})}\to {𝐻}^{𝑚({\top }^{𝑛})}$ is a continuous linear map satisfying

Show that the nullspace of $𝑃$ is a finite dimensional subspace of ${𝐻}^{𝑠({\top }^{𝑛})}$, and its range is closed in ${𝐻}^{𝑚({\top }^{𝑛})}$.