querent

Prove that a closed linear subspace of a reflexive Banach space is reflexive.

For $(𝑋,𝜇)$ a measure space with $𝜇(𝑋)<\infty$, show that if ${𝑓}_{𝑖}\to 𝑓$ in measure and ${sup}_{𝑖}{\Vert {𝑓}_{𝑖}\Vert }_{{𝐿}^{𝑝}}<\infty$ for some $𝑝>1$ then ${𝑓}_{𝑖}\to 𝑓$ in ${𝐿}^1$.

Suppose $𝑓\in {\mathrm{𝒟︀}}^{\prime }(ℝ)$, with ${\mathrm{𝒟︀}}^{\prime }(ℝ)$ denoting the space of distributions on $ℝ$.

(a) Show that there exists $𝑢\in {\mathrm{𝒟︀}}^{\prime }(ℝ)$ such that ${𝑢}^{\prime }=𝑓$.

(b) Show that if ${𝑣}^{\prime }=𝑓$ as well, then $𝑢-𝑣$ is a distribution given by a constant function.

Suppose that $𝑓:[0,\infty )\to [0,\infty )$ is continuous and for any $𝑥$ in $[0,\infty )$ the sequence $𝑓(𝑥),𝑓(2𝑥),𝑓(3𝑥),\dots$ tends to zero. Show that ${lim}_{𝑥\to \infty }𝑓(𝑥)=0$.

For $𝑠\ge 0$ define ${𝐻}^{𝑠({ℝ}^{𝑛})}$ to be the subspace of ${𝐿}^2({ℝ}^{𝑛})$ consisting of $𝑓\in {𝐿}^2({ℝ}^{𝑛})$ with

and let ${\Vert 𝑓\Vert }_{{𝐻}^{𝑠({ℝ}^{𝑛})}}≔{\Vert {(1+{|𝜉|}^2)}^{𝑠/2}\widehat{𝑓}(𝜉)\Vert }_{{𝐿}^2({ℝ}^{𝑛})}$, where $\widehat{𝑓}$ denotes the Fourier transform of $𝑓$.

(a) Prove that there is no continuous map

so that $𝑃(𝑢,𝑣)=𝑢𝑣$ for $𝑢,𝑣\in {𝐶}_0^{\infty ({ℝ}^{𝑛})}$ (compactly supported smooth functions).

(b) On the other hand, for $𝑠>𝑛/2$, show that there is a continuous map

so that $𝑃(𝑢,𝑣)=𝑢𝑣$ for $𝑢,𝑣\in {𝐶}_0^{\infty ({ℝ}^{𝑛})}$.

Hint: First prove that for $𝑝>0$, there is $𝐶=𝐶(𝑝)$ so that

for any $𝜉,{𝜉}^{\prime }\in {ℝ}^{𝑛}$.

(a) Show that for each $𝐿>0$ there exists ${𝐶}_{𝐿}$ so that if $𝑓\in {𝐶}_{𝑐}^{\infty (ℝ)}$ and the support of $𝑓$ is contained inside the interval $[-𝐿,𝐿]$ then

(b) Assume that an inequality of the form ${\Vert 𝑓\Vert }_{{𝐿}^2}\le 𝐶{\Vert 𝑓\Vert }_{{𝐿}^1}^{𝑎}{\Vert \nabla 𝑓\Vert }_{{𝐿}^2}^{𝑏}$ holds for all $𝑓$ in the Schwartz class $\mathrm{𝒮︀}({ℝ}^{𝑛})$. Find the only possible values of $𝑎$ and $𝑏$ — note that they depend on the dimension $𝑛$.

(c) Use the Plancherel identity to show that if $\widehat{𝑓}(𝜉)=0$ for $|𝜉|\le 𝑅$ then ${\Vert 𝑓\Vert }_{{𝐿}^2}^2\le \frac{{𝐶}_1}{{𝑅}^2}{\Vert \nabla 𝑓\Vert }_{{𝐿}^2}^2$, and that if $\widehat{𝑓}(𝜉)=0$ for $|𝜉|\ge 𝑅$, then ${\Vert 𝑓\Vert }_{{𝐿}^2}^2\le {𝐶}_1{𝑅}^{𝑛}{\Vert 𝑓\Vert }_{{𝐿}^1}^2$, with a constant ${𝐶}_1$ that depends only on the dimension $𝑛$. Combine these estimates to prove an inequality of the form ${\Vert 𝑓\Vert }_{{𝐿}^2}\le 𝐶{\Vert 𝑓\Vert }_{{𝐿}^1}^{𝑎}{\Vert \nabla 𝑓\Vert }_{{𝐿}^2}^{𝑏}$ with $𝑎$ and $𝑏$ you have found in part (b), and a constant $𝐶$ that depends only on the dimension $𝑛$.

Two short problems:

(a) Suppose $(𝑋,\mathrm{ℬ︀},𝜇)$ is a measure space and $𝑓$ is a non-negative measurable function. Show that

(b) Suppose that $𝑓\in {𝐿}^1(ℝ)$. Show that the Fourier transform of $𝑓$, given by $\widehat{𝑓}(𝜉)={\int }_{ℝ}{𝑒}^{-𝑖𝑥𝜉}𝑓(𝑥)d𝑥$, $𝜉\in ℝ$, is continuous and ${lim}_{|𝜉|\to \infty }\widehat{𝑓}(𝜉)=0$.

Suppose that $𝑋$ is a Banach space.

(a) Show that the natural inclusion $𝜄:𝑋\to {𝑋}^{\ast \ast }$, given by $(𝜄(𝑥))(𝜆)=𝜆(𝑥)$, $𝑥\in 𝑋$, $𝜆\in {𝑋}^{\ast }$, is an isometry.

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

(a) Suppose $𝑋$ is a Banach space, ${𝐴}_{𝑛}\in \mathrm{ℒ︀}(𝑋)$ are compact operators, $𝑛\in \mathbb{N}$. Suppose ${\{{𝐴}_{𝑛}\}}_{𝑛\in \mathbb{N}}$ converges to $𝐴\in \mathrm{ℒ︀}(𝑋)$ in the operator norm topology. Show that $𝐴$ is compact.

(b) Give (with proof) an example of a sequence ${\{{𝐴}_{𝑛}\}}_{𝑛\in \mathbb{N}}$ of compact operators on a Banach space $𝑋$ converging to a non-compact operator $𝐴\in \mathrm{ℒ︀}(𝑋)$ in the strong operator topology.

Let $𝜇$ be the Lebesgue measure on ${ℝ}^{𝑛}$, ${𝐵}_{𝑟(𝑥)}=\{𝑦\in {ℝ}^{𝑛}:|𝑦-𝑥|<𝑟\}$. For $𝑓\in {𝐿}_{loc}^1({ℝ}^{𝑛})$, define the maximal function

(a) Show that there is $𝐶>0$ (depending only on $𝑛$) so that if $𝑓\in {𝐿}^1({ℝ}^{𝑛})$ then

(b) For any non-zero $𝑓\in {𝐿}^1({ℝ}^{𝑛})$, show that $𝑀𝑓$ is not in ${𝐿}^1({ℝ}^{𝑛})$.