querent

Let $(𝑋,𝐴,𝜇)$ be a measure space and let ${\{{𝑓}_{𝑛}\}}_{𝑛=1}^{\infty }$ be a sequence of nonnegative functions in ${𝐿}^1(𝑋,𝜇)$. Let $𝑓\in {𝐿}^1(𝑋,𝜇)$. If

pointwise almost everywhere on $𝑋$, show that

Consider ${𝐿}^2([0,1])$. Define ${ℎ}_0,\{{ℎ}_{𝑛𝑗}\}$ as follows: let ${ℎ}_0={𝜒}_{[0,1]}$ (the indicator function of $[0,1]$) and for each $𝑛=1,2,\dots$, we divide the interval $[0,1]$ into $2^{𝑛-1}$ equal intervals ${𝐼}_{𝑗}$, for each $𝑗$, let ${𝐼}_{𝑗}^{+}$ and ${𝐼}_{𝑗}^{-}$ be the right and left halves of the interval ${𝐼}_{𝑗}$, then ${ℎ}_{𝑛𝑗}=2^{(𝑛-1)/2}({𝜒}_{{𝐼}_{𝑗}^{+}}-{𝜒}_{{𝐼}_{𝑗}^{-}})$. Let $𝑓\in {𝐿}^2([0,1])$ and ${𝑎}_0={\int }_0^1𝑓$, ${𝑎}_{𝑛𝑗}={\int }_0^1{ℎ}_{𝑛𝑗}𝑓$. Prove that

Let $𝑋,𝑌$ be Banach spaces over $ℝ$. Suppose $𝑇:𝑋\to 𝑌$ is a linear compact operator. Prove that ${𝑇}^{\ast }:{𝑌}^{\ast }\to {𝑋}^{\ast }$ is also compact. (Hint: Arzelà–Ascoli.)

Let $𝑓\in {𝐿}^1(ℝ)$ satisfy

Show that $𝑓=0$ almost everywhere.

Let $𝐻$ be a separable Hilbert space. We say that a bounded linear operator $𝑇:𝐻\to 𝐻$ is a shift if there exists an orthonormal basis ${\{{𝑒}_{𝑛}\}}_{𝑛=1}^{\infty }$ of $𝐻$ such that $𝑇({𝑒}_{𝑛})={𝑒}_{𝑛+1}$ for every $𝑛\in {\mathbb{Z}}_{\ge 1}$.

(a) If $𝑇$ is a shift, find its spectrum.

(b) Prove that a linear operator $𝑇:𝐻\to 𝐻$ is a shift if and only if $𝑇$ is a Hilbert space isomorphism (i.e., for all $𝑥,𝑦\in 𝐻$) satisfying ${\bigcap }_{𝑛=1}^{\infty }Ran({𝑇}^{𝑛})=\{0\}$ and ${dim(Ran(𝑇))}^{⟂}=1$.

(a) Give an example of Hilbert spaces $𝑋,𝑌$ and an operator $𝐴\in \mathrm{ℒ︀}(𝑋,𝑌)$ such that $Ran𝐴$ is not closed.

(b) Show that if $𝑋,𝑌$ are Hilbert spaces, $𝐴\in \mathrm{ℒ︀}(𝑋,𝑌)$, and $Ran𝐴$ is closed then $Ran{𝐴}^{\ast }$ is closed (where ${𝐴}^{\ast }\in \mathrm{ℒ︀}(𝑌,𝑋)$ is the Hilbert space adjoint). (Hint: Reduce to the case when $𝐴$ is invertible.)

(c) Show that if there exists $𝐶>0$ such that for all $𝑦\in 𝑌$, ${\Vert 𝑦\Vert }_{𝑌}\le 𝐶{\Vert {𝐴}^{\ast }𝑦\Vert }_{𝑋}$ then $𝐴$ is surjective. Is the conclusion still true if only ${𝐴}^{\ast }$ injective is assumed? Give a proof or provide a counterexample.

Prove that the function

is absolutely continuous on $(0,1)$.

(a) Prove that any tempered distribution $𝑢\in {\mathrm{𝒮︀}}^{\prime }({ℝ}^{𝑛})$ supported on $\{0\}$ takes the form

where $𝑁\in {\mathbb{Z}}_{\ge 0}$ and ${𝑐}_{𝛼}$ are constants.

(b) Let $𝑢\in {𝐶}^{\infty }({ℝ}^{𝑛})$ and suppose $𝜙\mapsto {\int }_{{ℝ}^{𝑛}}𝑢𝜙$ defines a tempered distribution. Assume that $\mathrm{\Delta }𝑢=0$ (where ). Prove that $𝑢$ is a polynomial. (Hint: Use the Fourier transform.)

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

(b) Prove that there exists ${𝐶}^{\prime }>0$ such that the following holds for all $1\le 𝑝\le \infty$ and all $𝑓\in \mathrm{𝒮︀}(ℝ)$ satisfying $supp(\mathrm{ℱ︀}𝑓)\subset 𝐵(0,1)$:

where $\mathrm{ℱ︀}$ denotes the Fourier transform. (Hint: Let $𝜒\in {𝐶}_{𝑐}^{\infty }(ℝ)$ satisfy $𝜒\equiv 1$ on $[-1,1]$ and $\equiv 0$ on $ℝ\setminus [-2,2]$. Write $\mathrm{ℱ︀}𝑓(𝜉)=𝜒(𝜉)\mathrm{ℱ︀}𝑓(𝜉)$.)

Let ${\{{𝑓}_{𝑛}\}}_{𝑛=1}^{\infty }\subset {𝐿}^{𝑝}(ℝ)$ and $𝑓\in {𝐿}^{𝑝}(ℝ)$ satisfy

and

(i) If $1<𝑝<\infty$, prove that

(ii) If $𝑝=1$, prove or disprove that