querent

Let $𝐴$ be a non-empty subset of ${\ell }^{\infty }(\mathbb{N})$. Define $𝑓:{\ell }^{\infty }(\mathbb{N})\to ℝ$ by

Prove that $𝑓$ is Lipschitz with constant $1$, that is, for any $𝑥,𝑦\in {\ell }^{\infty }(\mathbb{N})$,

Let $𝑓:[0,1]\to ℝ$ be non-decreasing. Recalling that this assumption implies that ${𝑓}^{\prime }(𝑥)$ exists for almost all $𝑥\in [0,1]$ (with respect to Lebesgue measure $𝑑𝑥$), prove that

Let $\mathrm{ℱ︀}$ denote the Fourier transform on ${ℝ}^{𝑛}$.

(a) Prove that there exists $𝐶>0$ such that the following holds for all $𝑓\in \mathrm{𝒮︀}({ℝ}^{𝑛})$:

(b) Prove that there exists $𝐶>0$ such that the following holds. For any Lebesgue measurable $𝐸\subset {ℝ}^{𝑛}$ with ${\mathrm{ℒ︀}}^{𝑛}(𝐸)<\infty$ and for all $𝑓\in \mathrm{𝒮︀}({ℝ}^{𝑛})$ with $supp(\mathrm{ℱ︀}𝑓)\subset 𝐸$:

If $𝑈\subset {ℝ}^{𝑛}$ is a bounded open set and $𝛿>0$, prove that there is a countable collection of closed balls such that ${𝜌}_{𝑖}\in (0,𝛿)$ for all $𝑖$,

whenever $𝑖\ne 𝑗$ and , where ${\mathrm{ℒ︀}}^{𝑛}$ denotes the Lebesgue measure.

Let $𝐸\subset ℝ$ be a bounded set, and let $𝑓:𝐸\to ℝ$ be a uniformly continuous function. Prove that $𝑓$ is bounded on $𝐸$.

Suppose that $(𝑋,𝐵,𝜇)$ is a measure space and $𝑓:𝑋\to ℝ$ and $𝑔:𝑋\to ℝ$ are integrable. For each $𝑡\in ℝ$, set

and

Part a: Assume that $(𝑋,𝐵,𝜇)$ is $𝜎$-finite. Prove that

Here, for subsets $𝐶$ and $𝐷$ of a set $𝑌$,

denotes their symmetric difference.

Part b: Show that the conclusion of part (a) holds even if $(𝑋,𝐵,𝜇)$ is not $𝜎$-finite.

Let ${𝐿}^0([0,1])$ be the vector space of Lebesgue measurable functions. Let $𝑑$ be the metric on ${𝐿}^0([0,1])$ given by

(a) Prove that ${𝑓}_{𝑛}\to 𝑓$ in the metric $𝑑$ if and only if ${𝑓}_{𝑛}\to 𝑓$ in measure.

(b) Let $\mathrm{𝒰︀}\subset {𝐿}^0([0,1])$ be a non-empty open convex neighborhood of $0$. Prove that $\mathrm{𝒰︀}={𝐿}^0([0,1])$.

(c) Suppose $𝑇:({𝐿}^0([0,1]),𝑑)\to ℝ$ is a continuous linear function. Prove that $𝑇$ is the zero map.

Let $\top =ℝ/(2𝜋\mathbb{Z})$. Given $𝑡\in ℝ$ and any Borel measure $𝜇$ on $\top$, define its translate ${𝜇}_{𝑡}$ by ${𝜇}_{𝑡}(𝐴)=𝜇({𝐴}_{𝑡})$, where ${𝐴}_{𝑡}=\{𝑥:𝑥+𝑡\in 𝐴\}$ for any Borel set $𝐴$.

(a) Is the map $𝑡\mapsto {𝜇}_{𝑡}$ necessarily continuous in the topology given by the dual norm, where measures are viewed as the dual of $𝐶(\top )$? Prove or disprove.

(b) Is the map $𝑡\mapsto {𝜇}_{𝑡}$ necessarily continuous in the weak-* topology on measures as the dual of $𝐶(\top )$? Prove or disprove.

(c) Is the map $𝑡\mapsto {𝜇}_{𝑡}(𝐴)$ necessarily continuous when $𝐴$ is Borel? Prove or disprove.

(d) Is the map $𝑡\mapsto {𝜇}_{𝑡}(𝐴)$ necessarily continuous if we in addition assume that $𝜇$ is absolutely continuous with respect to the Lebesgue measure? Prove or disprove.

Let $𝑋$ be a compact metric space and let $𝑓:𝑋\to ℝ$ and ${𝑓}_{𝑛}:𝑋\to ℝ$ be continuous functions for $𝑛\in \mathbb{N}$. Assume that ${𝑓}_{𝑛}\le {𝑓}_{𝑛+1}$ for all $𝑛\in \mathbb{N}$ and that ${𝑓}_{𝑛}$ converges pointwise to $𝑓$. Show that the convergence is uniform and that this claim is not true in general if $𝑓$ is not assumed to be continuous.

For $𝑥\in ℝ$ and $𝑡>0$, let

and

Part a: Let $𝑢$ be a continuous function on $ℝ$ that vanishes outside of a compact set. Prove that the functions

converge uniformly to $𝑢$ on $ℝ$ as $𝑡\to \infty$.

Part b: Let $𝑢\in {𝐿}^{𝑝}(ℝ)$ for some $𝑝\in [1,\infty )$. Prove that the functions

converge to $𝑢$ in ${𝐿}^{𝑝}(ℝ)$ as $𝑡\to \infty$.