querent

Show that there is no norm on ${𝐶}^0([0,1])$ such that convergence with respect to this norm is the same as pointwise convergence.

Does the integral

converge? Justify your answer.

Assume that $𝑓\in {𝐿}^1([0,1])$. Compute

Justify your answer.

(a) Suppose $𝑋,𝑍$ are reflexive Banach spaces, $𝑗:𝑋\to 𝑍$ a continuous injection (so one can consider $𝑋$ a subspace of $𝑍$) with dense range. Show that ${𝑗}^{\ast }:{𝑍}^{\ast }\to {𝑋}^{\ast }$ is also a continuous injection with a dense range.

(b) Suppose now $𝑉,𝑍,{𝑋}_1,{𝑋}_2$ are Hilbert spaces with $𝑉\subset {𝑋}_{𝑖}\subset 𝑍$ for $𝑖=1,2$ where all inclusions are continuous injections with dense range. Equip ${𝑋}_1\cap {𝑋}_2\subset 𝑍$ with the norm ${\Vert 𝑧\Vert }_{{𝑋}_1}+{\Vert 𝑧\Vert }_{{𝑋}_2}$. Show that for every $𝜆\in {({𝑋}_1\cap {𝑋}_2)}^{\ast }$, there exist ${𝜇}_1\in {𝑋}_1^{\ast }$ and ${𝜇}_2\in {𝑋}_2^{\ast }$ (both identified with subspaces of ${𝑉}^{\ast }$) such that $𝜆={𝜇}_1{|}_{{𝑋}_1\cap {𝑋}_2}+{𝜇}_2{|}_{{𝑋}_1\cap {𝑋}_2}$.

Let $𝐻$ be a separable Hilbert space. Suppose ${𝐴}_{𝑛},𝐴:𝐻\to 𝐻$ are bounded linear operators such that ${𝐴}_{𝑛}\to 𝐴$ in the strong operator topology.

(a) Show that ${\Vert {𝐴}_{𝑛}\Vert }_{\mathrm{ℒ︀}(𝐻,𝐻)}$ is bounded.

(b) Suppose that $𝐾:𝐻\to 𝐻$ is a compact linear operator. Show that ${𝐴}_{𝑛}𝐾\to 𝐴𝐾$ in operator norm.

(a) Show that if $𝑀$ is a compact connected metric space, and $𝑓:𝑀\to ℝ$ is continuous, then the image $𝑓(𝑀)\subset ℝ$ is a finite closed interval.

(b) Suppose $𝑀$ is a metric space such that for any continuous function $𝑓:𝑀\to ℝ$, the image $𝑓(𝑀)\subset ℝ$ is a finite closed interval. Show that $𝑀$ must be compact and connected.

Let $𝐸\subset ℝ$ be uncountable. Prove that there exists ${𝑥}_0\in 𝐸$ such that, for every $𝛿>0$,

is uncountable.

Let $𝑓:ℝ\to [0,\infty )$ be a measurable function supported on $[0,1]$. For each $𝑘\in {\mathbb{Z}}_{\ge 0}$, define

where ${\mathrm{𝟙}}_{[2^{-𝑘}𝑚,2^{-𝑘}(𝑚+1)]}$ denotes the indicator function of the interval $[2^{-𝑘}𝑚,2^{-𝑘}(𝑚+1)]$. Prove that for any $0\le 𝑘\le \ell$ integers,

In this problem, $\top =ℝ/\mathbb{Z}$ and ${𝐻}^2(\top )=\{𝑓\in {𝐿}^2(\top )|{\Vert 𝑓\Vert }_{{𝐻}^2(\top )}<\infty \}$, where ${\Vert 𝑓\Vert }_{{𝐻}^2(\top )}^2={\sum }_{𝑘\in \mathbb{Z}}{(1+{|𝑘|}^2)}^2{|\mathrm{ℱ︀}(𝑓)|}^2(𝑘)$, and $\mathrm{ℱ︀}$ denotes the Fourier series map.

(a) Let $𝑇:{𝐶}^{\infty }(\top )\to {𝐶}^{\infty }(\top )$ be the linear operator given by $𝑇(𝑓)=-{𝑓}^{″}+𝑓$. Prove that $𝑇$ extends to an operator $\widetilde{𝑇}:{𝐻}^2(\top )\to {𝐿}^2(\top )$ and that $\widetilde{𝑇}$ is invertible.

(b) Prove that there exists $𝐶>0$ such that the following holds for all $𝑓\in {𝐻}^2(\top )$:

(c) Prove that there exists an $𝜖>0$ such that

has a solution $𝑓\in {𝐻}^2(\top )$ whenever ${\Vert 𝑔\Vert }_{{𝐿}^2(\top )}\le 𝜖$. (Hint: Define $𝑆:{𝐻}^2(\top )\to {𝐻}^2(\top )$ by $𝑆(𝑓)={\widetilde{𝑇}}^{-1}({𝑓}^3+𝑔)$. To show that $𝑆$ has a fixed point, it suffices by the contraction mapping principle to show that there exists a closed subset $\mathrm{ℬ︀}\subset {𝐻}^2(\top )$ such that $𝑆(\mathrm{ℬ︀})\subset \mathrm{ℬ︀}$ and that there exists $𝑟\in (0,1)$ such that ${\Vert 𝑆({𝑓}_1)-𝑆({𝑓}_2)\Vert }_{{𝐻}^2(\top )}\le 𝑟{\Vert {𝑓}_1-{𝑓}_2\Vert }_{{𝐻}^2(\top )}$ whenever ${𝑓}_1,{𝑓}_2\in \mathrm{ℬ︀}$.)

Show that for any continuously differentiable function $𝑓:[0,1]\to ℝ$ and any $𝜀>0$, there exists a polynomial $𝑝(𝑥)$ such that

and