querent

Suppose that $𝑋,𝑌$ are reflexive separable Banach spaces, ${𝑋}^{\ast },{𝑌}^{\ast }$ the duals, $𝑃\in \mathrm{ℒ︀}(𝑋,𝑌)$, and suppose that its adjoint ${𝑃}^{\ast }\in \mathrm{ℒ︀}({𝑌}^{\ast },{𝑋}^{\ast })$ satisfies the following property: There is a Banach space $𝑍$ and a compact map $𝜄:{𝑌}^{\ast }\to 𝑍$ and $𝐶>0$ such that for all $𝜑\in {𝑌}^{\ast }$,

(a) Show that $Ker{𝑃}^{\ast }$ is finite dimensional.

(b) Let $𝑉$ be a closed subspace of ${𝑌}^{\ast }$ with $𝑉\oplus Ker{𝑃}^{\ast }={𝑌}^{\ast }$. Show that there is ${𝐶}^{\prime }>0$ such that for $𝜑\in 𝑉$, ${\Vert 𝜑\Vert }_{{𝑌}^{\ast }}\le {𝐶}^{\prime }{\Vert {𝑃}^{\ast }𝜑\Vert }_{{𝑋}^{\ast }}$.

(c) Show that for $𝑓\in 𝑌$ such that $\ell (𝑓)=0$ for all $\ell \in Ker{𝑃}^{\ast }$, there is $𝑢\in 𝑋$ such that $𝑃𝑢=𝑓$.

Suppose that $(𝑋,\mathrm{𝒜︀},𝜇)$ is a measure space, with $𝜇(𝑋)=1$, $𝑓$ a measurable function on $𝑋$ such that $𝑓>0$ $𝜇$-almost everywhere. Show that for all $𝜀>0$ there is $𝛿>0$ such that $𝐴\in \mathrm{𝒜︀}$, $𝜇(𝐴)\ge 𝜀$ implies ${\int }_{𝐴}𝑓d𝜇\ge 𝛿$.

Prove that there is a signed Radon measure $𝜈$ on $[0,1]$ such that ${\int }_{[0,1]}𝑝d𝜈={𝑝}^{\prime }(0)$ for every polynomial of degree at most $100$. Is there a signed Radon measure $𝜇$ such that ${\int }_{[0,1]}𝑝d𝜇={𝑝}^{\prime }(0)$ for every polynomial $𝑝$? Prove or disprove.

Consider ${𝑐}_0\subseteq {\ell }^{\infty }$, where ${𝑐}_0$ is the space of real valued sequences converging to $0$, equipped with the supremum norm (and ${\ell }^{\infty }$ is the space of bounded sequences, with the sup norm, as usual).

(a) Is ${𝑐}_0\subseteq {\ell }^{\infty }$ dense in the weak topology? Prove or disprove.

(b) Viewing ${\ell }^{\infty }={({\ell }^1)}^{\ast }$, we can endow ${\ell }^{\infty }$ with the weak-* topology. Is ${𝑐}_0\subseteq {\ell }^{\infty }$ dense in the weak-* topology? Again, prove or disprove.

Suppose $𝐻$ is an infinite dimensional separable Hilbert space.

(a) Give an example of a bounded operator, $𝐴\in \mathrm{ℒ︀}(𝐻)$, such that $𝐴$ is non-compact and ${𝐴}^2=0$.

(b) Show that if $𝐾\in \mathrm{ℒ︀}(𝐻)$ is compact, $𝐾={𝐾}^{\ast }$ and ${𝐾}^{𝑛}=0$ for some $𝑛\ge 2$ then $𝐾=0$.

(c) Suppose $𝐴\in \mathrm{ℒ︀}(𝐻)$ is bounded, $𝐴{𝐴}^{\ast }={𝐴}^{\ast }𝐴$ and ${𝐴}^{𝑛}=0$ for some $𝑛\ge 2$. Does it follow that $𝐴=0$? Prove or disprove.

(a) Let

for $𝜑\in \mathrm{𝒮︀}(ℝ)$. Show that $𝐺$ is a well-defined tempered distribution.

(b) Define what one means by "the Fourier transform of $𝐺$" and compute it.

Suppose that $𝐾\in {𝐿}^{𝑝([0,1]\times [0,1])}$, $1<𝑝<\infty$. Let $𝑞$ be the dual exponent, ${𝑝}^{-1}+{𝑞}^{-1}=1$.

(a) For $𝑓\in {𝐿}^{𝑞([0,1])}$, let $(𝐴𝑓)(𝑥)=\int 𝐾(𝑥,𝑦)𝑓(𝑦)d𝑦$. Show that $(𝐴𝑓)(𝑥)$ indeed exists for almost every $𝑥$ and $𝐴\in \mathrm{ℒ︀}({𝐿}^{𝑞([0,1])},{𝐿}^{𝑝([0,1])})$.

(b) Suppose that for every $𝑓\in {𝐿}^{𝑞([0,1])}$, $(𝐴𝑓)(𝑥)=0$ for almost every $𝑥$. Show that $𝐾=0$ a.e.

Recall that for $𝑠\ge 0$ the Sobolev space ${𝐻}^{𝑠({ℝ}^{𝑛})}$ consists of $𝑢\in {𝐿}^2({ℝ}^{𝑛})$ with

where $\mathrm{ℱ︀}$ is the Fourier transform. Let ${ℝ}^{𝑛}={ℝ}^{𝑛-𝑘}\oplus {ℝ}^{𝑘}$, $0<𝑘\le 𝑛$, be the standard decomposition (corresponding to the standard basis). Let $𝑅:\mathrm{𝒮︀}({ℝ}^{𝑛})\to \mathrm{𝒮︀}({ℝ}^{𝑛-𝑘})$ denote the restriction map $(𝑅𝑢)({𝑥}^{\prime })=𝑢({𝑥}^{\prime },0)$, ${𝑥}^{\prime }\in {ℝ}^{𝑛-𝑘}$, where $({𝑥}^{\prime },0)={ℝ}^{𝑛-𝑘}\oplus {ℝ}^{𝑘}={ℝ}^{𝑛}$. Show that for $𝑠>𝑘/2$, $𝑅$ has a unique continuous extension to a map $𝑅:{𝐻}^{𝑠({ℝ}^{𝑛})}\to {𝐻}^{𝑠-𝑘/2}({ℝ}^{𝑛-𝑘})$.

Suppose that $\{{𝑎}_{𝑛}:𝑛=0,1,2,\dots \}$ is any sequence of real numbers. Show that there exists a real valued function $𝑓\in {𝐶}^{\infty (ℝ)}$ such that ${𝑓}^{(𝑛)}(0)={𝑎}_{𝑛}$.

Consider a metric space $(𝑋,𝑑)$. For $𝑆\subseteq (𝑋,𝑑)$ write $Lim(𝑆)$ for the set of limit points of $𝑆$, ${Lim}^2(𝑆)=Lim(Lim(𝑆))$, and ${Lim}^{𝑁(𝑆)}$ for the limit point operation iterated $𝑁$ times. Assume that $𝑆\subseteq (𝑋,𝑑)$ is compact and has ${Lim}^{𝑁(𝑆)}=\varnothing$ for some $𝑁$. Prove that for any $𝜀>0$ there is a finite set ${𝑥}_1,\dots ,{𝑥}_{𝑚}$ in $𝑆$ and ${𝑟}_{𝑗}<𝜀$ so that $𝑆\subseteq {\bigcup }_{𝑗=1}^{𝑚}{𝐵}_{{𝑟}_{𝑗}}({𝑥}_{𝑗})$ and the open balls ${𝐵}_{{𝑟}_{𝑗}}({𝑥}_{𝑗})$ are pairwise disjoint.