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We say that a sequence ${\{{𝑎}_{𝑛}\}}_{𝑛=1}^{\infty }$ in $[0,1]$ is equidistributed (in $[0,1]$) if for all intervals $[𝑐,𝑑]\subseteq [0,1]$, the proportion of elements ${𝑎}_{𝑛}\in [𝑐,𝑑]$ for $𝑛\le 𝑁$ converges as $𝑁\to \infty$ to $𝑑-𝑐$.

(a) Show that $\{{𝑎}_{𝑛}\}\subseteq [0,1]$ is equidistributed if and only if the measures ${𝜇}_{𝑁}=\frac{1}{𝑁}{\sum }_{1\le 𝑛\le 𝑁}{𝛿}_{{𝑎}_{𝑛}}$ (with ${𝛿}_{{𝑎}_{𝑛}}$ the unit mass at ${𝑎}_{𝑛}$) converge in the weak-* topology to the Lebesgue measure.

(b) Prove that the sequence ${𝑎}_{𝑛}=𝑛𝛼mod1$, $𝑛\ge 1$, is equidistributed if and only if $𝛼$ is irrational.

Let $𝜇$ denote the Lebesgue measure on $[0,1]$. Suppose that ${𝑓}_{𝑘}$, $𝑘\in \mathbb{N}$, are Lebesgue measurable (real-valued) functions on $[0,1]$, and let ${𝑚}_{𝑘𝑛}=𝜇(\{𝑥\in [0,1]:|{𝑓}_{𝑘}|\in (2^{𝑛-1},2^{𝑛}]\})$ for $𝑛\in \mathbb{N}$.

(a) Suppose that ${\sum }_{𝑛=1}^{\infty }𝑛2^{𝑛}{𝑚}_{𝑘𝑛}\le 1$ for all $𝑘$ and suppose that ${𝑓}_{𝑘}\to 0$ a.e. Show that $\int {𝑓}_{𝑘}d𝜇\to 0$.

(b) Give an example of ${𝑓}_{𝑘}$ as above such that ${\sum }_{𝑛=1}^{\infty }{2}^{𝑛}{𝑚}_{𝑘𝑛}\le 1$ for all $𝑘$, ${𝑓}_{𝑘}\to 0$ a.e, but $\int {𝑓}_{𝑘}d𝜇$ does not tend to 0.

Suppose $𝑋$ is a compact metric space, and $\ell$ is a positive linear functional on $𝐶(𝑋)$, the Banach space of real valued continuous functions, i.e. $𝑓\ge 0$ implies $\ell (𝑓)\ge 0$. Suppose $𝐾\subseteq 𝑋$ compact. Let ${𝑂}_{𝑗}$, $𝑗\in \mathbb{N}$, be open sets satisfying and ${\cap }_{𝑗\in \mathbb{N}}{𝑂}_{𝑗}=𝐾$, and let ${𝑓}_{𝑗}\in 𝐶(𝑋;[0,1])$ be such that ${𝑓}_{𝑗}=1$ on $𝐾$, ${𝑓}_{𝑗}=0$ on the complement of ${𝑂}_{𝑗}$. Show directly from first principles (without quoting a major theorem) that ${lim}_{𝑗\to \infty }\ell ({𝑓}_{𝑗})$ exists, and is independent of the choice of ${𝑂}_{𝑗}$ and ${𝑓}_{𝑗}$ satisfying these conditions.

Let ${\mathrm{𝒮︀}}^{\prime }({ℝ}^{𝑛})$ denote the space of tempered distributions on ${ℝ}^{𝑛}$. A locally ${𝐿}^1$ function $𝑓$ on ${ℝ}^{𝑛}$ (i.e. the restriction to compact sets is ${𝐿}^1$) is said to lie in ${\mathrm{𝒮︀}}^{\prime }({ℝ}^{𝑛})$ if the map ${𝐶}_{𝑐}^{\infty ({ℝ}^{𝑛})}\ni 𝜑\mapsto {\int }_{{ℝ}^{𝑛}}𝑓𝜑\in ℂ$ has a (necessarily unique) continuous extension to an element of ${\mathrm{𝒮︀}}^{\prime }({ℝ}^{𝑛})$. Here ${𝐶}_{𝑐}^{\infty ({ℝ}^{𝑛})}$ is the set of compactly supported ${𝐶}^{\infty }$ functions on ${ℝ}^{𝑛}$.

(a) Show that the function ${|𝑥|}^{-(𝑛-𝛽)}$ lies in the space ${\mathrm{𝒮︀}}^{\prime }({ℝ}^{𝑛})$ if $0<𝛽<𝑛$. For what values of $𝛽$ is it a sum of an ${𝐿}^1$ and a ${𝐿}^2$ function?

(b) Show that the function ${𝑒}^{𝑥}cos({𝑒}^{𝑥})$ on $ℝ$ lies in ${\mathrm{𝒮︀}}^{\prime }(ℝ)$.

(c) Prove that there is a locally ${𝐿}^1({ℝ}^{𝑛})$ function that does not lie in ${\mathrm{𝒮︀}}^{\prime }({ℝ}^{𝑛})$.

Suppose $𝑝\in (0,1)$, $0<𝑎<𝑝$, and suppose $𝑁\in \mathbb{N}$, ${𝐴}_1,\dots ,{𝐴}_{𝑁}$ are Lebesgue measurable subsets of $[0,1]$ with average measure $\frac{1}{𝑁}{\sum }_{𝑖=1}^{𝑁}𝜇({𝐴}_{𝑖})\ge 𝑝$. Let $𝐸=\{𝑥\in [0,1]:𝑥\in {𝐴}_{𝑖}\text{for at least}𝑎𝑁\text{values of}𝑖\}$.

(a) Show that $𝜇(𝐸)\ge \frac{𝑝-𝑎}{1-𝑎}$.

(b) Show that if $𝑐>\frac{𝑝-𝑎}{1-𝑎}$ then there exist $𝑁\in \mathbb{N}$ and sets ${𝐴}_1,\dots ,{𝐴}_{𝑁}$ with $𝜇({𝐴}_{𝑖})\ge 𝑝$ for all $𝑖$ such that $𝜇(𝐸)<𝑐$.

Suppose that $(𝑋,{\Vert \cdot \Vert }_{𝑋})$ is a normed vector space, $𝑀$ and $𝑁$ are (not necessarily closed) subspaces equipped with norms ${\Vert \cdot \Vert }_{𝑀}$, resp. ${\Vert \cdot \Vert }_{𝑁}$ such that the identity maps $(𝑀,{\Vert \cdot \Vert }_{𝑀})\to (𝑀,{\Vert \cdot \Vert }_{𝑋})$, resp. $(𝑁,{\Vert \cdot \Vert }_{𝑁})\to (𝑁,{\Vert \cdot \Vert }_{𝑋})$ are continuous. Let $𝑀+𝑁$ be the algebraic sum: $𝑀+𝑁=\{𝑚+𝑛\in 𝑋:𝑚\in 𝑀,𝑛\in 𝑁\}$. For $𝑥\in 𝑀+𝑁$, let

(a) Show that ${\Vert \cdot \Vert }_{𝑀+𝑁}$ is a norm on $𝑀+𝑁$.

(b) Show that if $(𝑀,{\Vert \cdot \Vert }_{𝑀})$ and $(𝑁,{\Vert \cdot \Vert }_{𝑁})$ are complete then $(𝑀+𝑁,{\Vert \cdot \Vert }_{𝑀+𝑁})$ is complete.

Let $\mathrm{𝒮︀}$ denote the set of Schwartz functions on ${ℝ}^{𝑛}$. The Fourier transform on $\mathrm{𝒮︀}$, and more generally on ${𝐿}^1$, is given by $(\mathrm{ℱ︀}𝜑)(𝜉)=\int {𝑒}^{-𝑖𝑥\cdot 𝜉}𝜑(𝑥)d𝑥$, while on tempered distributions $𝑣\in {\mathrm{𝒮︀}}^{\prime }$ it is given by $(\mathrm{ℱ︀}𝑣)(𝜑)=𝑣(\mathrm{ℱ︀}𝜑)$, $𝜑\in \mathrm{𝒮︀}$.

(a) Show that there exists a compactly supported ${𝐶}^{\infty }$ function $𝜑$ on $ℝ$ such that $𝜑\ge 0$, $𝜑(0)>0$, and $\mathrm{ℱ︀}𝜑$ is non-negative, $(\mathrm{ℱ︀}𝜑)(0)>0$. (Hint: when is the Fourier transform of a function real?)

(b) Show that if $𝑢$ is a compactly supported distribution then the Fourier transform (as defined above) is ${𝐶}^{\infty }$, and there exists $𝑁\in ℝ$ such that for all $𝛼\in {\mathbb{N}}^{𝑛}$ there exist ${𝐶}_{𝛼}>0$ with .

Suppose that $𝑓,𝑔$ are holomorphic functions on a non-empty open connected set $\mathrm{\Omega }\subseteq ℂ$, and suppose that ${|𝑓|}^2+{|𝑔|}^2$ is constant on $\mathrm{\Omega }$. Show that $𝑓$ and $𝑔$ are constant on $\mathrm{\Omega }$.

Suppose $𝑋$ is a vector space over $ℂ$ and $\mathrm{ℱ︀}$ is a collection of linear maps $𝑋\to ℂ$. Equip $𝑋$ with the $\mathrm{ℱ︀}$-weak topology, i.e. the weakest topology in which all elements of $\mathrm{ℱ︀}$ are continuous.

(a) Show that the vector space operations $+:𝑋\times 𝑋\to 𝑋$ and $\cdot :ℂ\times 𝑋\to 𝑋$ are continuous in this topology (where $𝑋\times 𝑋$ and $ℂ\times 𝑋$ are equipped with the product topology).

(b) Suppose that $𝜌:𝑋\to [0,\infty )$ is continuous and is a seminorm. Show that there exist $𝑘\in \mathbb{N}$, ${\ell }_1,\dots ,{\ell }_{𝑘}\in \mathrm{ℱ︀}$ and $𝐶>0$ such that $𝜌(𝑥)\le 𝐶{\sum }_{𝑗=1}^{𝑘}|{\ell }_{𝑗}(𝑥)|$ for all $𝑥\in 𝑋$.

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 ${𝑃}^{\prime }\in \mathrm{ℒ︀}({𝑌}^{\ast },{𝑋}^{\ast })$ 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 $Ker{𝑃}^{\prime }$ is finite dimensional.

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

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