querent

Does there exist $𝑓\in 𝐶([0,1])$ with ${\int }_0^1𝑥𝑓(𝑥)d𝑥=1$ and ${\int }_0^1{𝑥}^{𝑛}𝑓(𝑥)d𝑥=0$ for $𝑛=0,2,3,\dots$? Give an example or prove that no such $𝑓$ exists.

Let ${\{{𝑓}_{𝑛}(𝑥)\}}_{𝑛=1}^{\infty }$ be a sequence of continuous functions defined on $[𝑎,𝑏]$. Suppose

converges pointwise on $[𝑎,𝑏)$, but

diverges. Show that

does not converge uniformly on $[𝑎,𝑏)$.

Consider $𝑓\in {𝐿}_{loc}^1({ℝ}^{𝑛})$ and assume that the (uncentered) maximal function $𝑀𝑓$ (given by $𝑀𝑓(𝑥)={sup}_{𝐵\ni 𝑥}\frac{1}{|𝐵|}{\int }_{𝐵}|𝑓|$, where the supremum is taken over all balls containing $𝑥$) satisfies $𝑀𝑓>𝜆>0$ on ${𝐵}_1(0)$. Show that $𝑀𝑓>𝑐𝜆$ on ${𝐵}_2(0)$ for some $𝑐>0$ depending only on the dimension $𝑛$. (Here, ${𝐵}_{𝑅(0)}=\{𝑥\in {ℝ}^{𝑛}:|𝑥|<𝑅\}$.)

Let $𝑉$ be the vector space of all real-valued Borel measurable functions on $[0,1]$. Show that there is no seminorm $\Vert \cdot \Vert$ on $𝑉$ such that a sequence $\{{𝑓}_{𝑛}\}$ converges in (the Lebesgue) measure to $𝑓$ on $[0,1]$ if and only if ${lim}_{𝑛\to \infty }\Vert 𝑓-{𝑓}_{𝑛}\Vert =0$.

Hint: assume that such a seminorm exists and show that for each $𝜀>0$ there is a collection of functions ${𝑔}_1,\dots ,{𝑔}_{𝑛}$ in $𝑉$ with $\Vert {𝑔}_{𝑗}\Vert \le 𝜀$ and ${𝑛}^{-1}{\sum }_{𝑗}{𝑔}_{𝑗}=1$.

Let ${\{{𝑎}_{𝑛}\}}_{𝑛=1}^{\infty }$ be a bounded sequence of real numbers. Suppose that

Prove that

Suppose $𝑋,𝑌$ are Hilbert spaces, $𝑃:𝑋\to 𝑌$ continuous linear. Suppose $𝑍$ is a dense subspace of $𝑌$ and for each $𝑧\in 𝑍$ there exists $𝑥\in 𝑋$ such that $𝑃𝑥=𝑧$. Consider in addition the following two estimates: (i) (for all $𝑥\in 𝑋$) ${\Vert 𝑥\Vert }_{𝑋}\le 𝐶{\Vert 𝑃𝑥\Vert }_{𝑌}$, (ii) (for all $𝑣\in 𝑌$) ${\Vert 𝑣\Vert }_{𝑌}\le 𝐶{\Vert {𝑃}^{\ast }𝑣\Vert }_{𝑋}$. Does either one of these imply that $𝑃:𝑋\to 𝑌$ is invertible? For both inequalities either give a proof of invertibility or provide a counterexample.

Let $\ell \subseteq {\ell }^{\infty }$ be the space of convergent real sequences $𝑎={\{{𝑎}_{𝑛}\}}_{𝑛=1}^{\infty }$. Define ${𝑇}_0:\ell \to ℝ$ by ${𝑇}_0(𝑎)={lim}_{𝑛\to \infty }{𝑎}_{𝑛}$.

(a) Prove that there exists a linear functional $𝑇:{\ell }^{\infty }\to ℝ$ such that $𝑇={𝑇}_0$ on $\ell$ and for $𝑎={\{{𝑎}_{𝑛}\}}_{𝑛=1}^{\infty }$,

Hint: first ignore the inequality involving ${lim\; inf}_{𝑛\to \infty }{𝑎}_{𝑛}$.

(b) Fix a $𝑇$ as in part (a). For each $𝐸\subseteq \mathbb{N}$, define ${𝑎}_{𝐸}={\{{𝑎}_{𝐸,𝑛}\}}_{𝑛=1}^{\infty }$ by setting ${𝑎}_{𝐸,𝑛}=1$ when $𝑛\in 𝐸$ and ${𝑎}_{𝐸,𝑛}=0$ otherwise. Define the function $𝜅:2^{\mathbb{N}}\to ℝ$ by $𝜅(𝐸)=𝑇({𝑎}_{𝐸})$. Show that $𝜅$ is finitely additive but not countably additive.

Suppose a sequence of real-valued functions $\{{𝑓}_{𝑛}\}$, where ${𝑓}_{𝑛}:ℝ\to ℝ$, is given such that for all $𝑛\in \mathbb{N}$ we have ${𝑓}_{𝑛}(0)=0$, and for any $𝑥,𝑦\in ℝ$,

Prove that the sequence $\{{𝑓}_{𝑛}\}$ has a subsequence that converges pointwise on $ℝ$.

For every $𝑓\in \mathrm{𝒮︀}({ℝ}^2)$, define $𝑇𝑓(𝑥,𝑦,𝑧)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{𝑒}^{𝑖(𝑥{𝜉}_1+𝑦{𝜉}_2+𝑧|𝜉|)}𝑓({𝜉}_1,{𝜉}_2)d{𝜉}_{1}d{𝜉}_2$, where $|𝜉|=\sqrt{{|{𝜉}_1|}^2+{|{𝜉}_2|}^2}$.

(a) Prove that there exists $𝐶>0$ such that the following holds for all $𝑓\in \mathrm{𝒮︀}({ℝ}^2)$ and for all $𝑧\in ℝ$:

(b) Prove that there exists $𝐶>0$ such that the following holds for all $𝑓\in \mathrm{𝒮︀}({ℝ}^2)$ satisfying $supp(𝑓)\subseteq \{({𝜉}_1,{𝜉}_2):|𝜉|\le 2|{𝜉}_1|\}$ and for all $𝑥\in ℝ$:

(c) Prove that for any $(𝑥,𝑦)\in {ℝ}^2$ and every $𝑓\in \mathrm{𝒮︀}({ℝ}^2)$, there is a $𝐶>0$ (depending on $(𝑥,𝑦)$ and $𝑓$) such that the following holds for all $𝑧\in ℝ$:

Recall that for $𝑠\ge 0$ the Sobolev space ${𝐻}^{𝑠({ℝ}^{𝑛})}$ consists of $𝑢\in {𝐿}^2({ℝ}^{𝑛})$ with $\int {(1+{|𝜉|}^2)}^{𝑠}{|(\mathrm{ℱ︀}𝑢)(𝜉)|}^{2}d𝜉<\infty$, where $\mathrm{ℱ︀}$ is the Fourier transform.

(a) Prove that for every $𝑠>𝑛/2$, there exists a constant ${𝐶}_1>0$ (depending only on $𝑠$ and $𝑛$) such that for every $𝑓\in {𝐻}^{𝑠({ℝ}^{𝑛})}$,

(b) Prove that for every $𝑠>𝑛/2$, there exists a constant ${𝐶}_2>0$ (depending only on $𝑠$ and $𝑛$) such that for every $𝑓,𝑔\in {𝐻}^{𝑠({ℝ}^{𝑛})}$,

Hint: first prove that for every $𝑝>0$, there is $𝐶=𝐶(𝑝)$ so that

(c) Let $𝑠>𝑛/2$ and $𝑓\in {𝐻}^{𝑠({ℝ}^{𝑛})}$. Prove that there exists $𝜀>0$ such that if ${\Vert 𝑓\Vert }_{{𝐻}^{𝑠({ℝ}^{𝑛})}}\le 𝜀$, then $ℎ(𝑥)=\frac{1}{1-𝑓(𝑥)}-1$ is well-defined (i.e. $1-𝑓(𝑥)\ne 0$ for a.e. $𝑥$) and $ℎ\in {𝐻}^{𝑠({ℝ}^{𝑛})}$.