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Let

denote the closed ball with radius $𝑟$. Evaluate the following integral:

Recall that the volume of ${𝐵}_{𝑛}(𝑟)$ is

Suppose that $𝑋$ is a Banach space.

(a) Suppose that the weak and the weak-* topologies on ${𝑋}^{\ast }$ are the same. Show that $𝑋$ is reflexive.

(b) Show that every closed subspace of $𝑋$ (closed with respect to the norm topology) is weakly closed.

Two short problems:

(a) Suppose that $𝑓:ℝ\to ℝ$ is increasing, and is differentiable almost everywhere with respect to the Lebesgue measure. Show that ${\int }_0^1{𝑓}^{\prime }(𝑥)d𝑥\le 𝑓(1)-𝑓(0)$.

(b) Let $\mathrm{𝒮︀}(ℝ)$, resp. ${\mathrm{𝒮︀}}^{\prime }(ℝ)$, denote the set of Schwartz functions, resp. tempered distributions, on $ℝ$.

For $𝜑\in \mathrm{𝒮︀}(ℝ)$, $𝜀>0$, $𝑘\in \mathbb{Z}$, define ${𝑢}_{𝜀}:\mathrm{𝒮︀}(ℝ)\to ℂ$ by ${𝑢}_{𝜀(𝜑)}={\int }_{ℝ}{(𝑥+𝑖𝜀)}^{-𝑘}𝜑(𝑥)d𝑥$.

Show that for all $𝜀>0$, ${𝑢}_{𝜀}\in {\mathrm{𝒮︀}}^{\prime }(ℝ)$, and that there exists $𝑢\in {\mathrm{𝒮︀}}^{\prime }(ℝ)$ such that for all $𝜑\in \mathrm{𝒮︀}(ℝ)$, ${𝑢}_{𝜀(𝜑)}\to 𝑢(𝜑)$ as $𝜀\to 0$.

Suppose that $𝐾$ is a bounded measurable function on $\{(𝑥,𝑦):0\le 𝑦\le 𝑥\le 1\}\subset {[0,1]}^2$. Let $𝑇$ be the Volterra operator

$𝑓\in {𝐿}^{𝑝([0,1])}$, $1<𝑝<\infty$. Show that $𝑇$ is well-defined, and $𝑇:{𝐿}^{𝑝([0,1])}\to {𝐿}^{𝑝([0,1])}$ is bounded. Show moreover that the spectrum of $𝑇$ is $\{0\}$.

Suppose $(𝑋,\mathrm{𝒯︀})$ is a compact Hausdorff topological space, and ${𝑓}_{𝑗}$, $𝑗\in \mathbb{N}$, are real valued continuous functions on $𝑋$ that separate points, i.e. if $𝑥,𝑦\in 𝑋$, $𝑥\ne 𝑦$ then there exists $𝑗\in \mathbb{N}$ such that ${𝑓}_{𝑗(𝑥)}\ne {𝑓}_{𝑗(𝑦)}$. Show that $(𝑋,\mathrm{𝒯︀})$ is metrizable.

Let $\mathrm{𝒮︀}({ℝ}^{𝑛})$, resp. ${\mathrm{𝒮︀}}^{\prime }({ℝ}^{𝑛})$, denote the set of Schwartz functions, resp. tempered distributions, on ${ℝ}^{𝑛}$.

(a) Suppose that $𝑢\in {\mathrm{𝒮︀}}^{\prime }({ℝ}^{𝑛})$ and ${𝑥}_{𝑗}𝑢=0$ for $𝑗=1,\dots ,𝑛$. Show that there exists $𝑐\in ℂ$ such that $𝑢=𝑐{𝛿}_0$.

(b) Suppose that $𝑓\in {\mathrm{𝒮︀}}^{\prime }(ℝ)$, ${𝜓}_0\in \mathrm{𝒮︀}(ℝ)$ with ${\int }_{ℝ}{𝜓}_0(𝑥)d𝑥\ne 0$, and $𝑎\in ℝ$. Show that there is a unique $𝑢\in {\mathrm{𝒮︀}}^{\prime }(ℝ)$ such that ${𝑢}^{\prime }=𝑓$ and $𝑢({𝜓}_0)=𝑎$.

(a) Show that on $\top =ℝ/(2𝜋\mathbb{Z})$, $𝑓\in {𝐿}^{𝑝(\top )}$, $1\le 𝑝<\infty$ implies that ${\int }_{\top }{|𝑓(𝑥+ℎ)-𝑓(𝑥)|}^{𝑝}d𝑥\to 0$ as $ℎ\to 0$.

(b) Suppose that $𝑓\in {𝐿}^{𝑝(\top )}$, $1<𝑝<\infty$, and ${sup}_{ℎ\ne 0,|ℎ|<1}{\int }_{\top }{|(𝑓(𝑥+ℎ)-𝑓(𝑥))/ℎ|}^{𝑝}d𝑥<\infty$. Show that the distributional derivative of $𝑓$, ${𝑓}^{\prime }$, satisfies ${𝑓}^{\prime }\in {𝐿}^{𝑝(\top )}$.

A number $𝜉\in ℝ$ is diophantine (with exponent $𝑘$, $𝑘>0$) if for some constant $𝐶>0$ there are no rational numbers $𝑝/𝑞$ ($𝑝,𝑞\in \mathbb{Z}$, $𝑞>0$) such that $|𝜉-𝑝/𝑞|<𝐶{𝑞}^{-𝑘}$.

$𝜉$ is a Liouville number if it is not diophantine, i.e. if for every $𝑘>0$ there exist integers $𝑝,𝑞$ such that $𝑞>𝑘$ and $|𝜉-𝑝/𝑞|\le {𝑞}^{-𝑘}$.

(a) Prove that the set of diophantine numbers is of the first category in $ℝ$; equivalently: the set of Liouville numbers is residual in $ℝ$.

Hint: Write ${𝐸}_{𝑘}={\cap }_{𝑁}{\cup }_{𝑝,𝑞\in \mathbb{Z},𝑞>𝑁}\{𝜉:|𝜉-𝑝/𝑞|<{𝑞}^{-𝑘}\}$, and show that $\mathrm{ℒ︀}={\cap }_{𝑘}{𝐸}_{𝑘}$.

(b) Prove that the set $\mathrm{ℒ︀}$ of Liouville numbers has zero Hausdorff dimension. This means: for every $𝜀>0$ and finite interval $𝐼$ there are intervals ${𝐼}_{𝑛}$, $𝑛\in \mathbb{N}$, such that $𝐼\cap \mathrm{ℒ︀}\subset {\cup }_{𝑛\in \mathbb{N}}{𝐼}_{𝑛}$, and ${\sum }_{𝑛}{|{𝐼}_{𝑛}|}^{𝜀}<𝜀$.

Let ${𝐶}_{𝑐({ℝ}^{𝑁})}$ denote the set of continuous real-valued functions of compact support on ${ℝ}^{𝑁}$. Let ${𝜇}_{𝑛}$, $𝑛\in \mathbb{N}$, and $𝜇$ be locally finite Borel measures on ${ℝ}^{𝑁}$ such that $\int 𝑓d{𝜇}_{𝑛}\to \int 𝑓d𝜇$ for all $𝑓\in {𝐶}_{𝑐({ℝ}^{𝑁})}$.

(a) Show that if $𝑈\subset {ℝ}^{𝑁}$ is open then $𝜇(𝑈)\le {lim\; inf}_{𝑛\to \infty }{𝜇}_{𝑛(𝑈)}$.

(b) Show that if $𝑆$ is a bounded Borel set and if then $𝜇(𝑆)={lim}_{𝑛\to \infty }{𝜇}_{𝑛(𝑆)}$.

Suppose that $𝑃(𝜉)={\sum }_{|𝛼|\le 𝑚}{𝑎}_{𝛼}{𝜉}^{𝛼}$, ${𝑎}_{𝛼}\in ℂ$, is a polynomial of degree $𝑚$ on ${ℝ}^{𝑛}$; here for $𝛼\in {\mathbb{N}}^{𝑛}$, $|𝛼|={\sum }_{𝑗=1}^{𝑛}{𝛼}_{𝑗}$, and ${𝜉}^{𝛼}={𝜉}_1^{{𝛼}_1}\dots {𝜉}_{𝑛}^{{𝛼}_{𝑛}}$. Let $𝑃(𝐷)$ be the corresponding differential operator, $𝑃(𝐷)={\sum }_{|𝛼|\le 𝑚}{𝑎}_{𝛼}{𝐷}^{𝛼}$, , ${𝐷}^{𝛼}={𝐷}_1^{{𝛼}_1}\dots {𝐷}_{𝑛}^{{𝛼}_{𝑛}}$. We say that $𝑃$ is elliptic if ${ℝ}^{𝑛}\ni 𝜉\ne 0$ implies ${\sum }_{|𝛼|=𝑚}{𝑎}_{𝛼}{𝜉}^{𝛼}\ne 0$.

(a) Show that if $𝑃$ is elliptic, $𝑢\in {\mathrm{𝒮︀}}^{\prime }({ℝ}^{𝑛})$ and $𝑃𝑢\in \mathrm{𝒮︀}({ℝ}^{𝑛})$ then $𝑢\in {𝐶}^{\infty ({ℝ}^{𝑛})}$.

(b) Recall that for $𝑚\ge 0$, ${𝐻}^{𝑚({\top }^{𝑛})}$ is the subset of ${𝐿}^2({\top }^{𝑛})$ consisting of functions whose Fourier coefficients satisfy ${\sum }_{𝑘\in {\mathbb{Z}}^{𝑛}}{(1+{|𝑘|}^2)}^{𝑚}{\left|\widehat{𝑓}(𝑘)\right|}^2<\infty$. Here $\top =ℝ/(2𝜋\mathbb{Z})$ and $\widehat{𝑓}(𝑘)={(2𝜋)}^{-𝑛/2}\int {𝑒}^{-𝑖𝑥\cdot 𝑘}𝑓(𝑥)d𝑥$, $𝑘\in {\mathbb{Z}}^{𝑛}$.

Show that if $𝑃$ is elliptic of order $𝑚$ then $𝑃$ considered as an operator $𝑃:{𝐻}^{𝑚({\top }^{𝑛})}\to {𝐿}^2({\top }^{𝑛})$ has finite dimensional nullspace.