Show that the linear functional is unbounded in $C_{00}$. defined as $T$ is...
Given a linear functional $T: C_{00}\to C_{00}$. Where $C_{00}$ is space sequences with finitely many non-zero terms with $\ell_2$ norm.$T$ is defined as $$T(x)=\sum_{n=1}^{\infty}...
View ArticleProve that $\sup(A\cup B)=\max\{\sup(A),\sup(B)\}$
I want to prove $\sup(A\cup B)=\max\{\sup(A),\sup(B)\}$, where $A,B\subset \mathbb R$ are non-empty and bounded sets from above. I have reviewed similar questions and answers, but I intended to...
View ArticleTrying to prove monotone-sequences property ⇒ Archimedean property
Monotone-sequences property ⇒ Archimedean propertyToday I just started learning this and having trouble understanding parts of the proof. Sorry if these are easy to some, I just couldn't find this...
View ArticleCan I apply Jensen's Inequality in this Case?
Problem: I am interested in the following question in order to be to apply Jensen's inequality to prove that under the assumptions below, we...
View ArticleHow to visualize triangle using a software
How to visualize (using some plotting software) the triangle inequality, $|x-z|\le|x-y|+|y-z|$, where $x,y$ and $z$ belong to some finite intervals.I am asking for general way, for example,...
View ArticleEvery positive rational number $x$ can be expressed in the form $\sum_{k=1}^n...
I have this theorem which I can't prove.Please help."Show that every positive rational number $x$ can be expressed in the form $\sum_{k=1}^n \frac{a_k}{k!}$ in one and only one way where each $a_k$ is...
View ArticleWe get $L^2$ convergence, but do we get a.s. convergence?
Assume you have a sequence of independent Bernoulli random variables$X_i$ each with probability $p_i$. Let $c_i$ be a sequence of real numbers and $ m,M$ be a real numbers such that$0 < m...
View ArticleMeasure of the set $\{t: \sqrt2-t \in \mathbb Q^c\}$
Let $f$ be the characteristic function of irrationals in $[0,1]$. I want to compute estimate the integral $$\int_{\sqrt2-1}^{-1/2}f(\sqrt2-t)\ dt.$$I reduced the integral to the measure of the set...
View ArticleHow does this optimal x change with this parameter?
Consider an optimization problem$$\max_x V = f(x,a) + g(x,a)$$where $\frac{\partial^2 f}{\partial x^2},\frac{\partial^2 g}{\partial x^2} < 0$Let $x^*$ (the optimal $x$) be such that$$\frac{\partial...
View ArticleChoosing a random integer from all the natural numbers.
Context: I saw a comment that if we choose a random positive integer, no matter how big it is, it will be closer to $0$ than $\infty$. Although this statement is far from rigorous this made me wonder:...
View ArticleMonotony of functions [closed]
How can I prove that the next function $f:[0,1] \to \mathbb{R}$ defined as follows is not monotone for any subinterval $[a,b] \subseteq [0,1]$? Some suggestions?Let $(0,1)\cap\mathbb{Q} =...
View ArticleDoes there exist $\theta\in\mathbb{T}$ such that...
Suppose $\alpha$ is Diophantine, i.e., there exist $\gamma, \tau$ such that $\|n\alpha\|_{\mathbb T }\geq\frac{\gamma}{|n|^\tau}$. For any analytic function $V(\cdot)\in...
View ArticleDerivative of $f \in BV_{loc}(\mathbb{R})$ with $f =0$ a.e.
Suppose I have $f \in BV_{loc}(\mathbb{R})$ and some open set $K \subset \mathbb{R}$ with $f = 0$ a.e. on $K$. Then is it true that $\partial_{x}f$ must necessarily be a measure (and not $L^p$) if $f$...
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Onsager conjecture and properties of Besov Spaces
I am currently studying the result in Peter Constantin 1, Weinan E, and E. S. Titi, Onsager’s Conjecture on the Energy Conservation for Solutionsof Euler’s...
View ArticleClosed form for $\sum_{k=n}^{2n-1} k(-1)^k \sum_{i=k}^{2n-1} {2n \choose i+1}...
I've found this sum:$$S(n) = \sum_{k=n}^{2n-1} k(-1)^k \sum_{i=k}^{2n-1} {2n \choose i+1} {i \choose n}$$The inner sum is elliptical iirc, but perhaps the double sum has a nice expression. We can...
View ArticleWhy $\sin(nx)$ converges weakly in $L^2(-\pi,\pi)$?
Can anybody tell me why $\sin(nx)$ converges weakly in $L^2(-\pi,\pi)$. I can't see how $\sin(nx)$ can converge?Explanation with any other example will be nice as well.
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Help with this step in Calderon-Zygmund decomposition
I'm currently reading this pdf: https://uu.diva-portal.org/smash/get/diva2:1231351/FULLTEXT01.pdf On Singular Integral Operators, Marcus Vaktnäs. In the fourth line it says "so at least one of $\langle...
View ArticleCan any open set in $\mathbb{R}^d$ be countably union of closed sets
I've already know that $\{B(x,r):x\in\mathbb{Q}^d,r\in \mathbb{Q}\} $ is an countable base of $\mathbb{R}^d$. Intuitively, I wonder that can an open set $\Omega\subset \mathbb{R}^d$ be countably union...
View ArticleFind compactification of $\mathbb{R}$ which has a subset homeomorphic to...
Consider $X=\mathbb{R}$ with the standard topology.How can I find a compactification $Y$, such that $Y$ is (of course) compact, hausdorff and has a subset which is homeomorphic to $\mathbb{R}^2$?This...
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Issue in numerical integration of...
I am trying to numerically integrate the integral representation of $\operatorname{Ai}^2(x)$. The representation...
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