querent

Let $𝜑:ℝ\to ℂ$ be $2𝜋$-periodic and Hölder continuous with exponent $𝛼$. Let ${𝑐}_{𝑛}=\frac{1}{2𝜋}{\int }_{-𝜋}^{𝜋}𝜑(𝑥){𝑒}^{-𝑖𝑛𝑥}d𝑥$ denote the $𝑛$-th Fourier coefficient of $𝜑$.

(a) Show that there exists $𝐶>0$ such that

for each $ℎ\in ℝ$.

(b) Suppose now that $𝛼>1/2$. Show that ${\sum }_{𝑛=-\infty }^{\infty }|{𝑐}_{𝑛}|<\infty$.

Two short problems.

(a) Does the fact that ${lim}_{𝑎\to 0}{\int }_{𝑎}^1𝑓d𝑥$ exists guarantee that $𝑓\in {𝐿}^1[0,1]$? Prove this or give a counterexample.

(b) Suppose that $𝑓:ℝ\to ℝ$ and for each $𝑥\in ℝ$ there exists a quadratic polynomial ${𝑃}_{𝑥}$, ${𝑃}_{𝑥}(𝑦)={𝑎}_{𝑥}{(𝑦-𝑥)}^2+{𝑏}_{𝑥}(𝑦-𝑥)+{𝑐}_{𝑥}$, such that ${lim}_{𝑦\to 𝑥}{|𝑦-𝑥|}^{-2}|𝑓(𝑦)-{𝑃}_{𝑥}(𝑦)|=0$. Does this imply that $𝑓$ is twice differentiable? Prove this or give a counterexample.

Two short problems.

(a) Suppose that $𝑋$ is a Banach space. Show that every closed subspace of $𝑋$ (closed with respect to the norm topology) is weakly closed.

(b) Let $𝑋,𝑌,𝑍$ be Banach spaces, $𝑇:𝑋\to 𝑌$, $𝑆:𝑌\to 𝑍$ be linear maps. Suppose $𝑆\circ 𝑇$ is bounded and $𝑆$ is both bounded and injective. Show that $𝑇$ is bounded.

Let ${𝑇}_{𝑡}:{𝐿}^2(ℝ)\to {𝐿}^2(ℝ)$, $𝑡\in ℝ$, be the operator given by $({𝑇}_{𝑡}𝑓)(𝑥)=𝑓(𝑥-𝑡)$.

(a) Suppose that $\Vert {𝑇}_{𝑡}-{𝑇}_{𝑠}\Vert <2$, where the norm is that of $\mathrm{ℒ︀}({𝐿}^2(ℝ))$, the space of bounded operators on ${𝐿}^2(ℝ)$. Show that $𝑡=𝑠$.

(b) Give (with proof) a locally convex topology on $\mathrm{ℒ︀}({𝐿}^2(ℝ))$ in which the map $ℝ\ni 𝑡\mapsto {𝑇}_{𝑡}\in \mathrm{ℒ︀}({𝐿}^2(ℝ))$ is continuous.

Let ${\{{𝑓}_{𝑛}\}}_{𝑛\in \mathbb{N}}$ be a sequence of measurable functions on a measurable space $(𝐸,\mathrm{ℰ︀})$. Show that the following functions are also measurable: ${inf}_{𝑛}{𝑓}_{𝑛}$, ${sup}_{𝑛}{𝑓}_{𝑛}$, ${lim\; inf}_{𝑛}{𝑓}_{𝑛}$, ${lim\; sup}_{𝑛}{𝑓}_{𝑛}$. Show also that $\{𝑥\in 𝐸:{𝑓}_{𝑛}(𝑥)\text{converges as}𝑛\to \infty \}\in \mathrm{ℰ︀}$.

Let ${\ell }^{\infty }$, resp. ${\ell }^2$, denote the vector spaces of bounded, resp. square summable, complex valued sequences with the standard norms.

(a) Let ${\{{𝑎}_{𝑛}\}}_{𝑛\in \mathbb{N}}\in {\ell }^{\infty }$. Define $𝑇:{\ell }^2\to {\ell }^2$ by ${(𝑇𝑥)}_{𝑛}={𝑎}_{𝑛}{𝑥}_{𝑛}$. Find (with proof) $𝜎(𝑇)$ (the spectrum of $𝑇$).

(b) Let $𝐾$ be a closed bounded non-empty subset of $ℂ$. Show that there exist $𝐻$ a Hilbert space and $𝑇:𝐻\to 𝐻$ a bounded linear map such that $𝜎(𝑇)=𝐾$.

Let ${𝐶}^{\infty (\top )}$ denote the vector space of infinitely differentiable complex-valued functions on $\top =ℝ/(2𝜋\mathbb{Z})$, and let , $𝜑\in {𝐶}^{\infty (\top )}$, be the ${𝐶}^{𝑘}$ norm of $𝜑$. Let $\mathrm{𝒯︀}$ be the weakest topology on ${𝐶}^{\infty (\top )}$ in which the functions

$𝑘\ge 0$, $𝜓\in {𝐶}^{\infty (\top )}$, are continuous.

(a) Show that $\mathrm{𝒯︀}$ is metrizable, and write down an explicit metric giving rise to the topology $\mathrm{𝒯︀}$.

(b) Show that there exists no norm $\Vert \cdot \Vert$ on ${𝐶}^{\infty (\top )}$ such that $\mathrm{𝒯︀}$ is the topology given by the norm $\Vert \cdot \Vert$.

Let ${𝐻}^1([0,\infty ))$ (Sobolev space on the half line) denote the completion of ${𝐶}_{𝑐}^1([0,\infty ))$ (${𝐶}^1$ functions on $[0,\infty )$ which vanish outside a compact set, but not necessarily at $0$) in the norm ${\Vert 𝑓\Vert }_{{𝐻}^1([0,\infty ))}={\Vert 𝑓\Vert }_{{𝐿}^2([0,\infty ))}+{\Vert {𝑓}^{\prime }\Vert }_{{𝐿}^2([0,\infty ))}$, and ${𝐻}^1(ℝ)$ be the standard Sobolev space (consisting of functions in ${𝐿}^2(ℝ)$ whose Fourier transform satisfies $\int (1+{|𝜉|}^2){|\mathrm{ℱ︀}𝑓(𝜉)|}^{2}d𝜉<\infty$).

(a) Show that the restriction map $𝑅:{𝐶}_{𝑐}^1(ℝ)\to {𝐶}_{𝑐}^1([0,\infty ))$ extends to a continuous linear map (still denoted by $𝑅$) $𝑅:{𝐻}^1(ℝ)\to {𝐻}^1([0,\infty ))$.

(b) Show that there is a continuous linear map $𝐸:{𝐻}^1([0,\infty ))\to {𝐻}^1(ℝ)$ such that $𝑅𝐸=𝐼$ (identity map).

Hint: Consider a map of the form $𝐸𝑓(𝑥)=𝑓(𝑥)$ when $𝑥\ge 0$, $𝐸𝑓(𝑥)={\sum }_{𝑗=1}^{𝑘}{𝑎}_{𝑗}𝑓(-𝑗𝑥)$ when $𝑥<0$, where $𝑘$ and ${𝑎}_{𝑗}$ are appropriately chosen.

Suppose $𝑓\in {𝐶}^{\infty ({ℝ}^{𝑛})}$ complex-valued with $Im𝑓\ge 0$ and $𝐾\subseteq {ℝ}^{𝑛}$ is compact. Suppose that for all points $𝑥\in 𝐾$ with $Im𝑓(𝑥)=0$, the differential of $𝑓$ does not vanish at $𝑥$. Show that for all $𝑢\in {𝐶}^{\infty ({ℝ}^{𝑛})}$ with support in $𝐾$ and for all $𝑁\ge 0$ there is $𝐶>0$ such that

Suppose that $𝐴$ is a bounded operator on a Hilbert space $𝐻$ and $𝐾$ is a compact operator on $𝐻$.

(a) If $𝐴$ has closed range, finite dimensional kernel and infinite dimensional cokernel, then show that $𝐴+𝐾$ also has all the same properties.

(b) If $𝐴$ has closed range, infinite dimensional kernel, finite dimensional cokernel, then show that $𝐴+𝐾$ has the same properties.

(c) Is it true that if $𝐴$ has closed range but kernel and cokernel both infinite dimensional, then $𝐴+𝐾$ still has closed range? Prove or give a counterexample.