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Let ${𝑋}_{𝑗}$, $𝑗=1,2,\dots$, be real-valued, independent and identically distributed random variables, such that the probability density $𝑝(𝑥)$ of each ${𝑋}_{𝑗}$ is a Schwartz class function, and set ${𝑍}_{𝑛}=({𝑋}_1+\dots +{𝑋}_{𝑛})/𝑛$. Show that the probability distribution function of ${𝑍}_{𝑛}$ is ${𝑝}_{𝑍}(𝑥)=𝑛[𝑝\ast 𝑝\ast \dots \ast 𝑝](𝑛𝑥)$, and there exists a constant $𝑠$ such that for any Schwartz function $𝑔(𝑥)$ we have $𝔼(𝑔({𝑍}_{𝑛}))\to 𝑔(𝑠)$.

A word on notation: a random variable $𝑋$ is a measurable function on a measure space $\mathrm{\Omega }$ equipped with a (non-negative) measure $𝜇$ such that ${\int }_{\mathrm{\Omega }}d𝜇=1$. A Borel measurable function $𝑝(𝑥)$ is the probability density of $𝑋$ if for any Borel set $𝐵\subseteq ℝ$ we have $𝜇(𝜔\in \mathrm{\Omega }:𝑋(𝜔)\in 𝐵)={\int }_{𝐵}𝑝(𝑥)d𝑥$. The expected value of a random variable $𝑍$ is $𝔼(𝑍)={\int }_{ℝ}𝑥𝑝(𝑥)d𝑥$. Finally, random variables ${𝑋}_1,{𝑋}_2,\dots ,{𝑋}_{𝑛}$ are independent if for any collection of Borel sets ${𝐵}_1,\dots ,{𝐵}_{𝑛}$ we have $𝜇(𝜔\in \mathrm{\Omega }:{𝑋}_1(𝜔)\in {𝐵}_1,\dots ,{𝑋}_{𝑛(𝜔)}\in {𝐵}_{𝑛})=𝜇(𝜔\in \mathrm{\Omega }:{𝑋}_1(𝜔)\in {𝐵}_1)\times \dots \times 𝜇(𝜔\in \mathrm{\Omega }:{𝑋}_{𝑛(𝜔)}\in {𝐵}_{𝑛})$.

Two short problems:

(a) Show that there is a closed subset $𝐸$ of $[0,1]$ of positive Lebesgue measure and with empty interior.

(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 $0$.

Suppose $𝑓$ is a non-negative Lebesgue measurable function on $[0,1]$ such that $𝑓>0$ almost everywhere. Show that for any $𝜀>0$ there is $𝛿>0$ such that if $𝐸$ is a Lebesgue measurable subset of $[0,1]$ with measure $𝑚(𝐸)\ge 𝜀$, then ${\int }_{𝐸}𝑓(𝑥)d𝑥\ge 𝛿$.

Let $𝑆$ be a closed subspace of $𝐶[0,1]$ (with the sup norm). Suppose that $𝑓\in 𝑆$ implies that $𝑓$ is continuously differentiable. Prove that $𝑆$ is finite dimensional.

Let $𝑋$ be a non-zero Banach space and $𝑇\in \mathrm{ℒ︀}(𝑋)$. Let $𝜌(𝑇)$ be the resolvent set, $𝜎(𝑇)$ the spectrum of $𝑇$.

(a) Suppose ${\{{𝜆}_{𝑛}\}}_{𝑛=1}^{\infty }\subseteq 𝜌(𝑇)$ and $𝜆\in 𝜎(𝑇)$ such that ${𝜆}_{𝑛}\to 𝜆$. Prove that

(b) Using part (a), or otherwise, show that there exists $𝜆\in 𝜎(𝑇)$ such that $𝜆𝐼-𝑇$ is not bounded below, i.e. for every $𝑐>0$, there exists $𝑥\in 𝑋-\{0\}$ such that ${\Vert (𝜆𝐼-𝑇)𝑥\Vert }_{𝑋}\le 𝑐{\Vert 𝑥\Vert }_{𝑋}$.

Let $𝐾\in {𝐿}_{ℝ}^2({ℝ}^2)$, i.e. a real valued element of the ${𝐿}^2$-space on ${ℝ}^2$. Define $𝑇:{𝐿}_{ℝ(ℝ)}^2\to {𝐿}_{ℝ(ℝ)}^2$ by

(a) Prove that $𝑇:{𝐿}_{ℝ(ℝ)}^2\to {𝐿}_{ℝ(ℝ)}^2$ is bounded and moreover that $𝑇$ is compact.

(b) For $𝛼\in ℝ$ and $𝑔\in {𝐿}_{ℝ(ℝ)}^2$, consider the following equation where $𝑓\in {𝐿}_{ℝ(ℝ)}^2$ is an unknown:

Prove that there exists $𝜀>0$ (depending only on $𝐾$) such that if $|𝛼|<𝜀$, then (1) admits a unique solution $𝑓\in {𝐿}_{ℝ(ℝ)}^2$.

(c) Suppose that ${\int }_{ℝ}{\int }_{ℝ}ℎ(𝑥)𝐾(𝑥,𝑦)ℎ(𝑦)d𝑥d𝑦\ge 0$ for all $ℎ\in {𝐿}_{ℝ(ℝ)}^2$. Prove that for all $𝛼\ge 0$, (1) admits a unique solution $𝑓\in {𝐿}_{ℝ(ℝ)}^2$.

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 $𝑓(𝑥)={𝑒}^{-𝑖𝐴𝑥\cdot 𝑥/2}$. Show that $𝑓$ is a tempered distribution, and 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.

Let ${𝐿}_{\ast }^2={𝐿}^2((0,\infty ),{𝑥}^{-1}d𝑥)$, i.e. ${\Vert 𝑓\Vert }_{{𝐿}_{\ast }^2}^2={\int }_0^{\infty }{|𝑓(𝑥)|}^2\frac{d𝑥}{𝑥}$. For each $𝑠\in ℝ$ consider the statement: there exists $𝐶>0$ such that for all $𝑢\in {𝐶}_0^{\infty ((0,\infty ))}$, . Find, with proof, the values of $𝑠$ for which this statement holds.

Hint: rewrite . Let $𝑡=log𝑥$, and rewrite the estimate in terms of ${𝐿}^2(ℝ,d𝑡)$. Then use the Fourier transform.

(a) Let $𝑋$ be a Banach space and $𝐸\subseteq {𝑋}^{\ast }$ be a subspace of ${𝑋}^{\ast }$ which is closed in the weak-* topology. Suppose moreover that ${\cap }_{𝜆\in 𝐸}Ker(𝜆)=\{0\}$. Prove that $𝐸={𝑋}^{\ast }$.

(b) Let $𝑋,𝑌$ be Banach spaces, $𝑇\in \mathrm{ℒ︀}(𝑋,𝑌)$. Let ${𝑇}^{\prime }\in \mathrm{ℒ︀}({𝑌}^{\ast },{𝑋}^{\ast })$ be the adjoint defined by ${𝑇}^{\prime }(𝜆)(𝑥)=𝜆(𝑇(𝑥))$ for $𝑥\in 𝑋$, $𝜆\in {𝑌}^{\ast }$. Prove that $𝑇$ is injective if and only if $im({𝑇}^{\prime })\subseteq {𝑋}^{\ast }$ is dense in the weak-* topology.

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.