Question Regarding proof of limsupXn is a random variable
Let $(\Omega, \mathscr{F})$ be a measurable space. If $\{X_n\}$ is a sequence of random variables and I want to prove $\limsup_{n\to\infty}X_n$ is a random variable. This is to show that$$\forall...
View ArticleHow to make sure that this function is continuous at (0,0)? [duplicate]
$f(x,y)= \begin{cases} \frac{x^{2}y^{2}}{x^{3}+y^{3}}, x\ne -y \\0, x=-y \end{cases} $I got that $\lim_{(x,y) \to (0,0)}f(x,y)=\lim_{r \to 0}r\frac{cos^{2}\phi sin^{2}\phi}{cos^{3}\phi + sin^{3}\phi}$...
View ArticleA continuous function can be bounded from above by a smooth 'adapted' function?
Consider a continuous function $f:\mathbb{R}\to\mathbb{R}$. Is there a standard strategy that produces a function $g\in C^{\infty}(\mathbb{R},\mathbb{R})$, such that$f(t)\le g(t)$$\forall t$;$g(t)$ can...
View ArticleConvergence of Sum in a Hilbert Space
I am having some issues understanding what I think may be a pretty fundamental fact about bases in a Hilbert space. If I have a basis $\{f_k\}_{k \geq 1}$ for the Hilbert space $L^2(0, 1)$, is it true...
View Articlemaximal monotone operator in $H_{0}^{1}(\Omega)$
I am considering an operator\begin{align}A:H_{0}^{1}(\Omega)&\to H^{-1}(\Omega)\\u&\mapsto \int_{\Omega}\nabla u\cdot\nabla v {dx}\mbox{ for all }v\in H_{0}^{1}(\Omega)\end{align}I already...
View ArticleHow do we rigorously eliminate $r^n$ and $\log r$ terms in a Fourier series...
In my PDE module, the general solution to Laplace's equation $\nabla^2 T=0$ in the plane (in polar coordinates) was shown to be $$T(r,\theta)=A_0+B_0\log...
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Convergence of left continuous inverse
Let $f$ be a non-decreasing function. The left continuous inverse of $f$ is given by $f^{\leftarrow}(x) = \inf \{s \colon f(s) \geq x \}$In the book Extreme value theory: an introduction, there's the...
View ArticleConditions for Convexity of a Product of Functions
I am studying the convexity of a product of functions and would like some assistance identifying additional conditions on one of the functions.Let $ f: \mathbb{R} \rightarrow \mathbb{R}^+ $ be a...
View ArticleSuppose if $\alpha =\sup S$ and $\alpha \notin S$ then $\alpha$ is an...
Suppose that $\alpha =\sup S$ and $\alpha \notin S$. Then prove that $\alpha$ is an accumulation point of $S$ and $S$ is infinite. I know thatLet $S\subseteq R$ be nonempty and bounded above, and let...
View ArticleProve that $\sum\sqrt{\dfrac{a^4+kb^2c^2}{a^2+kbc}} \geq a+b+c,\forall k...
I found a problem on this link: https://artofproblemsolving.com/community/c6h260824p1418804For $a,b,c > 0$. Prove that: $$\sum\sqrt{\dfrac{a^4+kb^2c^2}{a^2+kbc}} \geq a+b+c,\forall k...
View ArticleDo function in $u\in C^{1}_c(\mathbb{R}^n)\cap \dot{H}^1$ satisfy the...
Let $u\in C^{1}_c(\mathbb{R}^n)\cap \dot{H}^1$ be a smooth compactly supported function. Then is the following decay estimate correct?\begin{align}|u|(r,\theta) &\leq \int_r^\infty |\partial_{\tau}...
View ArticleShowing the existence of a limit
Please show me the existence of the limit clearly$$\lim_{\large(h,k)\to (0,0)}\dfrac{\vert hk\vert ^{\alpha} \log(h^2+k^2)}{\sqrt {h^2+k^2}} =0,$$for $\alpha > \frac12$.
View ArticleEssential Supremum on a closed interval
I'm currently studying real analysis and the topic of essential supremum came up. My book did not mention anything beyond a definition. I tried looking through online resources but most just give the...
View ArticleDecreasing sequence of open sets all containing $p\in{X}$ with infinite...
Suppose $\{U_n\}_{n=1}^{\infty}$ is a decreasing ($U_{n+1}\subset{U_n}, \forall{n}$) sequence of open sets in ametric space $(X,d)$ such that $\bigcap_{n=1}^{\infty}{U_n}=\{p\}$ for some $p\in{X}$....
View ArticleSome properties of averaged modulus of smoothness $\tau_{k}(f, \delta)_{p}$
I am currently reviewing a paper published in the Journal of Approximation Theory and working on proving some unresolved properties. Below are the necessary definitions and the properties...
View ArticleConvergence to Dirichlet function is not uniform
Let $r_{1},r_{2},...$ a sequence that includes all rational numbers in $[0,1]$. Define $$f_n(x)=\begin{cases}1&\text{if }x=r_{1},r_{2},...r_{n}\\0&\text{otherwise}\end{cases}$$this sequence...
View ArticleExample of a continuous function s.t. $f(\overline{A}) \subsetneq...
This question is a subproblem of the A map is continuous if and only if for every set, the image of closure is contained in the closure of image. I am not able to come up with any example of a...
View ArticleWhat is an explicit formula for $TT^*$ if $T:H\to\ell^2$ is the analysis...
Let $\{e_{n}\}_{n \in \mathbb{N}}$ be an orthonormal sequence for a Hilbert space $H$, let $ T: H \rightarrow \ell^{2}$ be the analysis operator $ T x=\left\{\left\langle x,...
View ArticleEvaluating $\lim_{x\to \infty} \frac{f^{-1}(2021x)-f^{-1}(x)}{\sqrt[2021]...
Let $f(x)=2021x^{2021}+x+1$, and compute the following limit:$$\lim_{x\to \infty} \frac{f^{-1}(2021x)-f^{-1}(x)}{\sqrt[2021] x}$$My attempt: i want to use mean value theorem to $f^{-1}(x)$ then we...
View ArticleIf $f_1 \in C_c (\mathbb{R}^N)$, Why is the set $\overline{B(0,1)} +...
I'm studying Brezis book: Functional Analysis, Sobolev spaces and partial differential equations.Theorem 4.22 states that if $f \in L^p(\mathbb{R}^N)$, with $1 \leq p < \infty$ and if $\rho_n$ is a...
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