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For $𝑠\ge 0$, let ${𝐻}^{𝑠(\top )}$ be the space of ${𝐿}^2$ functions $𝑓$ on the circle $\top =ℝ/(2𝜋\mathbb{Z})$ whose Fourier coefficients ${\widehat{𝑓}}_{𝑛}=\int {𝑒}^{-𝑖𝑛𝑥}𝑓(𝑥)d𝑥$ satisfy $\sum {(1+{𝑛}^2)}^{𝑠}{\left|{\widehat{𝑓}}_{𝑛}\right|}^2<\infty$, with norm ${\Vert 𝑓\Vert }_{𝑠}^2={(2𝜋)}^{-1}\sum {(1+{𝑛}^2)}^{𝑠}{\left|{\widehat{𝑓}}_{𝑛}\right|}^2$.

(a) Show that for $𝑟>𝑠\ge 0$, the inclusion map $𝜄:{𝐻}^{𝑟(\top )}\to {𝐻}^{𝑠(\top )}$ is compact.

(b) Show that if $𝑠>1/2$, then ${𝐻}^{𝑠(\top )}$ includes continuously into $𝐶(\top )$, and indeed the inclusion map is compact.

Two short problems.

(a) Show that if $𝑋$ is a Banach space and ${𝑋}^{\ast }={𝑋}^{\ast \ast \ast }$ (under the natural inclusion) then $𝑋={𝑋}^{\ast \ast }$.

(b) Show that if ${𝑥}_0\in {ℝ}^{𝑛}$ and $𝜀>0$ then there exists $𝜑\in {𝐶}_{𝑐}^{\infty ({ℝ}^{𝑛})}$ (compactly supported infinitely differentiable function) such that $𝜑({𝑥}_0)=1$ and $supp𝜑\subseteq \{𝑥\in {ℝ}^{𝑛}:|𝑥-{𝑥}_0|<𝜀\}$.

Suppose $𝐹$, ${𝐹}_{𝑛}$, $𝑛\ge 1$ integer, are increasing functions from the interval $[𝑎,𝑏]$, $𝑎<𝑏$, to $ℝ$ such that for all $𝑥\in [𝑎,𝑏]$, $𝐹(𝑥)={\sum }_{𝑛=1}^{\infty }{𝐹}_{𝑛}(𝑥)$. Prove that ${𝐹}^{\prime }(𝑥)={\sum }_{𝑛=1}^{\infty }{𝐹}_{{𝑛}^{\prime }}(𝑥)$ almost everywhere with respect to the Lebesgue measure.

(a) Suppose $𝑓,𝑔$ are positive measurable functions on $[0,1]$ and $𝑓(𝑥)𝑔(𝑥)\ge 1$ for $𝑥\in [0,1]$. Show that

(b) Suppose that $(𝑋,\mathrm{ℬ︀},𝜇)$ is a $𝜎$-finite measure space, $𝐾$ is a measurable function on $𝑋\times 𝑋$, and $\int |𝐾(𝑥,𝑦)|d𝜇(𝑦)\le 𝐶$, $\int |𝐾(𝑥,𝑦)|d𝜇(𝑥)\le 𝐶$ $𝜇$-a.e. Show that the integral operator $𝐴:{𝐿}^2(𝑋)\to {𝐿}^2(𝑋)$ defined by $(𝐴𝑓)(𝑥)=\int 𝐾(𝑥,𝑦)𝑓(𝑦)d𝜇(𝑦)$ is well-defined and bounded, and its norm is bounded by $𝐶$.

Suppose that $1<𝑝<\infty$, $𝑓,{𝑓}_{𝑛}\in {𝐿}^{𝑝([0,1])}$, $𝑛\in \mathbb{N}$, ${\Vert {𝑓}_{𝑛}\Vert }_{{𝐿}^{𝑝}}\le 1$ for all $𝑛$, and ${𝑓}_{𝑛}\to 𝑓$ almost everywhere. Show that ${𝑓}_{𝑛}\to 𝑓$ weakly and ${\Vert 𝑓\Vert }_{{𝐿}^{𝑝}}\le 1$.

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 ${𝜌}_{𝛼}(𝑥)\ne 0$, 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 $𝑥,𝑦\in 𝑋$,

Suppose $𝑋$ is a separable Hilbert space.

(a) Suppose $𝑇\in \mathrm{ℒ︀}(𝑋)$ is compact and ${𝑇}^{\ast }=𝑇$. Show that there is a complete orthonormal set in $𝑋$ consisting of eigenvectors of $𝑇$.

(b) Give an example (with proof) of a non-selfadjoint $𝑇\in \mathrm{ℒ︀}(𝑋)$ which is compact and which is such that the spectrum of $𝑇$ is $\{0\}$ but $𝑇$ has no eigenvectors.

Suppose $𝑢$ is a distribution (an element of the dual of ${𝐶}^{\infty }$) on the circle $\top =ℝ/(2𝜋\mathbb{Z})$. Show that there exists a function $𝑓\in 𝐶(\top )$, $𝑘\ge 0$ integer and $𝑐\in ℂ$ such that $𝑢=\frac{d^{𝑘}}{d{𝑥}^{𝑘}}𝑓+𝑐$, where $\frac{d^{𝑘}}{d{𝑥}^{𝑘}}$ is the $𝑘$th distributional derivative. (As usual, $𝐶(\top )$ is regarded as a subset of the set ${\mathrm{𝒟︀}}^{\prime }(\top )$ of distributions.)

Let $𝑋$ be an uncountable set equipped with the discrete topology. Let $\widehat{𝑋}$ be the one point compactification of $𝑋$, and let $𝐶\left(\widehat{𝑋}\right)$ be the Banach space of real-valued continuous functions on $\widehat{𝑋}$.

(a) Find (with proof) the $𝜎$-algebra of Baire sets (generated by compact ${𝐺}_{𝛿}$ sets) and the $𝜎$-algebra of Borel sets (generated by open sets).

(b) Find a $𝜎$-subalgebra $\mathrm{ℬ︀}$ of the Borel sets which contains the Baire sets and two distinct finite measures ${𝜇}_1,{𝜇}_2$ on $\mathrm{ℬ︀}$ such that $\int 𝑓d{𝜇}_1=\int 𝑓d{𝜇}_2$ for all $𝑓\in 𝐶\left(\widehat{𝑋}\right)$. Explain why the existence of these does not contradict the Riesz representation theorem concerning the dual of $𝐶\left(\widehat{𝑋}\right)$.

For each of the following maps $𝑓:ℝ\to 𝑋$, where $𝑋$ is a topological vector space, prove or disprove that the map is continuous, respectively differentiable. Here differentiability is the existence, for all $𝑡\in ℝ$, of the limit ${lim}_{ℎ\to 0}\frac{𝑓(𝑡+ℎ)-𝑓(𝑡)}{ℎ}$ in the space $𝑋$. We write $𝑓(𝑡)={𝑓}_{𝑡}$ below.

(a) $𝑋={𝐿}^2(ℝ)$, with standard norm, and ${𝑓}_{𝑡}(𝑥)={𝜒}_{[𝑡,𝑡+1]}(𝑥)$, ${𝜒}_{[𝑡,𝑡+1]}$ the characteristic (or indicator) function of $[𝑡,𝑡+1]$.

(b) $𝑋={𝐿}^2(ℝ)$, with standard norm, and ${𝑓}_{𝑡}(𝑥)=sin(𝑥-𝑡)$ if $𝑡\le 𝑥\le 𝑡+𝜋$, ${𝑓}_{𝑡}(𝑥)=0$ otherwise.

(c) $𝑋={\mathrm{𝒮︀}}^{\prime }(ℝ)$ (tempered distributions, the dual of Schwartz functions, $\mathrm{𝒮︀}(ℝ)$), with the weak-* topology, and ${𝑓}_{𝑡}={𝛿}_{𝑡}$, the delta distribution at $𝑡$.