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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 of order $𝑚$ if ${ℝ}^{𝑛}\ni 𝜉\ne 0$ implies ${\sum }_{|𝛼|=𝑚}{𝑎}_{𝛼}{𝜉}^{𝛼}\ne 0$. Suppose that $𝑃$ is elliptic of order $𝑚$.

Recall also 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 $\widehat{𝑓}(𝑘)={(2𝜋)}^{-𝑛/2}\int {𝑒}^{-𝑖𝑥\cdot 𝑘}𝑓(𝑥)d𝑥$, $𝑘\in {\mathbb{Z}}^{𝑛}$.

(a) Show that with $𝑃$ considered as a map $𝑃:{𝐻}^{𝑚({\top }^{𝑛})}\to {𝐿}^2({\top }^{𝑛})$, the nullspace of $𝑃$ is finite dimensional and is a subset of ${𝐶}^{\infty ({\top }^{𝑛})}$.

(b) Show that $𝑃$ is invertible as such a map if and only if it is injective.

Recall that for any $0<𝛼<1$, the space ${𝐶}^{𝛼([0,1])}$ is the set of continuous functions on $[0,1]$ with

equipped with the norm ${\Vert \cdot \Vert }_{{𝐶}^{𝛼}}$.

(a) Show that the unit ball of ${𝐶}^{𝛼([0,1])}$ has compact closure in $𝐶([0,1])$.

(b) Show that ${𝐶}^{𝛼([0,1])}$ is of first category in $𝐶([0,1])$.

Let $(𝑋,\Vert \cdot \Vert )$ be a Banach space.

(a) Show that every closed subspace of $𝑋$ is weakly closed.

(b) Show that if $𝑋$ is infinite dimensional then the weak closure of the unit sphere, $𝑆=\{𝑥\in 𝑋:\Vert 𝑥\Vert =1\}$, is the entire unit ball $𝐵=\{𝑥\in 𝑋:\Vert 𝑥\Vert \le 1\}$.

Two short problems.

(a) Show that there exists a continuous function $𝑓:[0,1]\to ℝ$ and a Lebesgue measurable set $𝐴\subseteq [0,1]$ with measure $𝑚(𝐴)=0$ such that $𝑓(𝐴)$ is measurable but has measure $𝑚(𝑓(𝐴))>0$.

(b) Show that if $𝑓:[0,1]\to ℝ$ is absolutely continuous and $𝐴$ is Lebesgue measurable with measure $0$ then $𝑓(𝐴)$ is measurable with measure $0$.

Two short problems.

(a) Suppose ${𝑓}_{𝑛}$ is a sequence in ${𝐿}^2([0,1])$ with ${\Vert {𝑓}_{𝑛}\Vert }_{{𝐿}^2}\le 1$ for all $𝑛$, and $𝑓\in {𝐿}^2([0,1])$. Consider the following two statements: (i) ${𝑓}_{𝑛}\to 𝑓$ in ${𝐿}^2$, (ii) ${𝑓}_{𝑛}(𝑥)\to 𝑓(𝑥)$ for almost every $𝑥\in [0,1]$. For each of the implications (i) $\Rightarrow$ (ii) and (ii) $\Rightarrow$ (i) either give a proof or give a counterexample.

(b) Show that there are Banach spaces $𝑋,𝑌$ and operators $𝐴,{𝐴}_{𝑛}\in \mathrm{ℒ︀}(𝑋,𝑌)$, $𝑛\in \mathbb{N}$, such that for every $𝑛$, $Ran{𝐴}_{𝑛}=𝐷$, and ${𝐴}_{𝑛}\to 𝐴$ in the strong operator topology, but $𝐷$ is a proper subspace of $Ran𝐴$. Can such an example exist if $𝐷$ is closed in $𝑌$?

Suppose $𝑋,𝑌$ are Hilbert spaces and $𝐴\in \mathrm{ℒ︀}(𝑋,𝑌)$, the vector space of all bounded linear operators from $𝑋$ to $𝑌$.

(a) Show that there is a unique operator ${𝐴}^{\ast }:𝑌\to 𝑋$ such that for all $𝑥\in 𝑋$ and $𝑦\in 𝑌$. Show further that ${𝐴}^{\ast }\in \mathrm{ℒ︀}(𝑌,𝑋)$ with ${\Vert {𝐴}^{\ast }\Vert }_{\mathrm{ℒ︀}(𝑌,𝑋)}={\Vert 𝐴\Vert }_{\mathrm{ℒ︀}(𝑋,𝑌)}$.

(b) Show that $𝐴$ has closed range if and only if ${𝐴}^{\ast }\in \mathrm{ℒ︀}(𝑌,𝑋)$ has closed range.

(a) Suppose that $𝑓\in {𝐿}^1(ℝ)$, $𝑓\ge 0$, and that $𝑓\ne 0$ as an element of ${𝐿}^1$. Let $\widehat{𝑓}(𝜉)={\int }_{ℝ}{𝑒}^{-𝑖𝑥\cdot 𝜉}𝑓(𝑥)d𝑥$ be the Fourier transform of $𝑓$. Show that $sup\left|\widehat{𝑓}\right|$ is attained exactly at $0$.

(b) Suppose $𝑓\in 𝑆(ℝ)$, $𝑓\ge 0$, $\int 𝑓(𝑥)d𝑥=1$, $\int 𝑥𝑓(𝑥)d𝑥=0$, and let ${𝑓}_1=𝑓$, ${𝑓}_{𝑘}={𝑓}_{𝑘-1}\ast 𝑓$, $𝑘\ge 2$. Show that with ${𝑔}_{𝑘(𝑥)}=𝑘{𝑓}_{𝑘(𝑘𝑥)}$, ${𝑔}_{𝑘}\to {𝛿}_0$ in distributions as $𝑘\to \infty$. (You may use part (a) even if you have not proved it.)

Let $𝑓\in 𝑆(ℝ)$ be a Schwartz function (i.e. $𝑓\in {𝐶}^{\infty (ℝ)}$ and is bounded for all $𝑗,𝑘\in {\mathbb{N}}_0$). Suppose too that ${\int }_{ℝ}{|𝑓(𝑥)|}^{2}d𝑥=1$. Recall the Fourier transform of $𝑓$,

Show that

Hint: Use Plancherel's theorem and the Cauchy-Schwarz inequality. The main point is to provide a positive lower bound which is independent of $𝑓$; if you cannot obtain the constant $1/4$, try to find some lower bound with a positive constant $𝑐$.

Let $\top =ℝ/\mathbb{Z}$, and let $𝜇$ be the Lebesgue measure on it. For $𝐴\subseteq \top$ and $𝑦\in \top$ write $𝐴+𝑦=\{𝑥+𝑦\in \top :𝑥\in 𝐴\}$. Suppose $𝐴\subseteq \top$ is measurable, and for each $𝑛\in {\mathbb{N}}^{+}$, $𝐴+2^{-𝑛}=𝐴$ (i.e. $𝐴$ is invariant under dyadic translations). Show that either $𝜇(𝐴)=0$ or $𝜇(𝐴)=1$.

Let $𝑋$ be a Banach space over $ℂ$ and $𝑀$ and $𝑁$ closed subspaces of $𝑋$. Write $𝑀+𝑁=\{𝑥\in 𝑋:\exists 𝑚\in 𝑀,𝑛\in 𝑁,𝑥=𝑚+𝑛\}$.

(a) Show that $𝑀+𝑁$ is closed if and only if there exists $𝐶>0$ such that for all $𝑥\in 𝑀+𝑁$ there exist $𝑚\in 𝑀$, $𝑛\in 𝑁$ such that $𝑥=𝑚+𝑛$ and $\Vert 𝑚\Vert +\Vert 𝑛\Vert \le 𝐶\Vert 𝑥\Vert$.

(b) Suppose that ${\ell }_{𝑀}:𝑀\to ℂ$ and ${\ell }_{𝑁}:𝑁\to ℂ$ are continuous linear functionals and ${\ell }_{𝑀}{|}_{𝑀\cap 𝑁}={\ell }_{𝑁}{|}_{𝑀\cap 𝑁}$. Show that if $𝑀+𝑁$ is closed, then there exists $\ell \in {𝑋}^{\ast }$ such that $\ell {|}_{𝑀}={\ell }_{𝑀}$ and $\ell {|}_{𝑁}={\ell }_{𝑁}$.

(c) Give an example of a Banach space $𝑋$ and closed subspaces $𝑀,𝑁$ such that $𝑀\cap 𝑁=\{0\}$ but $𝑀+𝑁$ is not closed.