querent

Recall that a topological space $𝑆$ is first countable if at each point $𝑝\in 𝑆$ there is a countable base for the topology, i.e. there exist ${𝑂}_{𝑛}$, $𝑛\in \mathbb{N}$, open such that $𝑂$ open and $𝑝\in 𝑂$ imply that for some $𝑛$, ${𝑂}_{𝑛}\subseteq 𝑂$.

(a) Suppose that $𝑋$ is a Banach space, ${\{{𝑥}_{𝑗}\}}_{𝑗=1}^{\infty }$ is a sequence in $𝑋$ with the property that ${\cup }_{𝑛=1}^{\infty }{𝑋}_{𝑛}=𝑋$, where ${𝑋}_{𝑛}=Span\{{𝑥}_1,\dots ,{𝑥}_{𝑛}\}$. Show that $𝑋$ is finite dimensional.

(b) Show that if a Banach space is first countable in its weak topology then it is finite dimensional. (Hint: show that the dual space has the property in part (a).)

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.

Recall that the Fourier transform of an ${𝐿}^1({ℝ}^{𝑛})$ function $𝑓$ is $(\mathrm{ℱ︀}𝑓)(𝜉)=\int {𝑒}^{-𝑖𝑥\cdot 𝜉}𝑓(𝑥)d𝑥$, where $\cdot$ is the standard inner product on ${ℝ}^{𝑛}$.

Let $𝐴$ be a real symmetric matrix, and define the function $𝑓$ by $𝑓(𝑥)={𝑒}^{-𝑖𝐴𝑥\cdot 𝑥/2}$, which is a tempered distribution. Find (with proof) its Fourier transform if $det(𝐴)\ne 0$. Make sure to give an explicit formula in terms of $𝐴$, without involving any limits. (Hint: Write $𝐴$ as the limit of complex symmetric matrices with negative definite imaginary part.)

Consider the collection of seminorms , $𝑘,𝑚\ge 0$ integer, on the subspace $𝑋$ of ${𝐶}^{\infty }$ functions on $[0,1]$ vanishing with all derivatives at 0.

(a) Show that the locally convex topology induced by these seminorms on $𝑋$ is the same as the restriction of the ${𝐶}^{\infty }$ topology to $𝑋$.

(b) Show that every $\{{𝜌}_{𝑘,𝑚}:𝑘,𝑚\ge 0\}$-continuous linear functional $\ell$ on $𝑋$ has an extension to a continuous linear functional $𝜆$ on ${𝐶}^{\infty ([-1,1])}$, with $𝑋$ identified with the subspace of ${𝐶}^{\infty ([-1,1])}$ consisting of functions vanishing on $[-1,0]$.

For $𝑘\in ℝ$, let ${\mathrm{\mathscr{H}︀}}_{+}^{𝑘}$ denote the set of distributions $𝑓$ on $\top =ℝ/(2𝜋\mathbb{Z})$ whose Fourier series $\widehat{𝑓}=\mathrm{ℱ︀}𝑓$ are rapidly decreasing as $𝑛\to -\infty$ (i.e. for all $𝑁$, $|\mathrm{ℱ︀}𝑓(𝑛)|\le {𝐶}_{𝑁}{|𝑛|}^{-𝑁}$ for $𝑛<0$), and which satisfy $\left|\widehat{𝑓}(𝑛)\right|\le {𝐶}_{𝑘}{(𝑛+1)}^{𝑘}$ for $𝑛\ge 0$, with the natural seminorms induced by these estimates. Show that there is $𝐾$ such that for $𝑓,𝑔\in {\mathrm{\mathscr{H}︀}}_{+}^{𝑘}$, $𝑓𝑔$ is well-defined in ${\mathrm{\mathscr{H}︀}}_{+}^{𝐾}$ (extending the usual multiplication on ${𝐶}^{\infty (\top )}$) and ${\mathrm{\mathscr{H}︀}}_{+}^{𝑘}\times {\mathrm{\mathscr{H}︀}}_{+}^{𝑘}\ni (𝑓,𝑔)\to 𝑓𝑔\in {\mathrm{\mathscr{H}︀}}_{+}^{𝐾}$ is continuous.

Let ${\ell }^2(\mathbb{Z})$ denote the Hilbert space of square summable bi-infinite complex valued sequences, and let $𝐿$ be the operator acting on ${\ell }^2(\mathbb{Z})$ defined by

($𝐿$ is called the discrete Laplacian.)

(a) Show that $𝐿$ is a bounded symmetric operator, with spectrum $[0,2]$, and find the eigenvalues of $𝐿$.

(b) Let $𝑉$ denote multiplication by a real-valued function $𝑣$: $(𝑉𝑓)(𝑛)=𝑣(𝑛)𝑓(𝑛)$, and suppose ${lim}_{|𝑛|\to \infty }𝑣(𝑛)=0$. Show that the spectrum of $𝐻=𝐿+𝑉$ outside $[0,2]$ is a discrete set.

Suppose that $𝐴$ is a Borel set in $ℝ$ with the property that if $𝑥\in 𝐴$ and if $𝑦$ is any other number such that the decimal expansions of $𝑥$ and $𝑦$ differ in at most finitely many places, then $𝑦\in 𝐴$. Prove that the Lebesgue measure of either $𝐴$ or $ℝ\setminus 𝐴$ is $0$.

Let $𝑓\in {𝐿}^1([0,1])$, and suppose that ${\int }_0^1𝑓{𝜑}^{(𝑛)}=0$ for every $𝜑\in {𝐶}_{𝑐}^{\infty (0,1)}$, where ${𝜑}^{(𝑛)}$ is the $𝑛$-th derivative. Show that $𝑓$ is a polynomial of degree at most $𝑛-1$.

Hint: Approximate $𝑓$ by smooth functions using convolutions.

Let $𝑇$ be a non-zero compact operator acting on a Hilbert space $𝐻$.

(a) Give an example, with proof, of such an operator such that $spec(𝑇)=\{0\}$.

(b) Prove that this is impossible if $𝑇$ is self-adjoint.

Let ${\ell }^2$ be the Hilbert space of square summable sequences, and $𝐻$ be the subspace consisting of sequences ${\{{𝑥}_{𝑛}\}}_{𝑛=1}^{\infty }$ with ${\sum }_{𝑛=1}^{\infty }{𝑛}^2{|{𝑥}_{𝑛}|}^2<\infty$. Show that $𝐻$ is of the first category in ${\ell }^2$.