Real part of inner product is greater than the smaller norm in closed convex...
Suppose $V$ is a Hilbert space, $U$ is is a nonempty closed convex subset of $V$, and $g ∈ U$ is the unique element of $U$ with smallest norm. Prove that $\operatorname{Re}\langle g,h\rangle ≥‖𝑔‖^2$,...
View ArticleThe set of all ellipsoids $\mathcal{E}(A)$ contained in a bounded open set...
We call $A$ a convex body if $A$ is a convex, non-empty, open, and bounded subset of $\Bbb R^d$. The open unit ball in $\Bbb R^d$ is denoted by $B_d$. In Tao-Vu's book, they say:Define an ellipsoid to...
View ArticlePointwise limit of continuous functions whose graph is in a given closed set
Let $C\subseteq\mathbb R^2$ be a closed set with the property that for every $x\in\mathbb R$, there exists at least one $y\in\mathbb R$ such that $(x,y)\in C$.Does there exist a function $f:\mathbb...
View Article$A =\{x\in\Bbb Q\mid 0≤x^2
$A =\{x\in\Bbb Q\mid 0≤x^2<2\}$. Is the closure of $A˚$ compact?I think that it is, because if I take the closure of the interior of A, I'm adding every limit point on A, for example sqrt(2), so the...
View ArticleCantor set Self-similarity
I’m defining the Cantor set in the following way$C:= \bigcap_{n\ge 0} C_n$ with $C_0:=[0,1]$ and $C_n:= \frac{1}{3} ( C_{n-1} \cup (C_{n-1}+2))$ for all $n\ge 1$I should prove that$T(C)=C$ with $T:...
View ArticleWhen is $ \sum_{k \in \mathbb{Z}}\left(\frac{\sin(k)}{k}\right)^{n}=2...
Define sequences $$a_n = \sum_{k \in \mathbb{Z}}\left(\dfrac{\sin(k)}{k}\right)^{n}, b_n = \int_{0}^{\infty}\left(\dfrac{\sin(x)}{x}\right)^{n}dx.$$I am trying to see if there is a relation between...
View ArticleIf $\psi=x\sin(\frac{1}{x})$, and $f$ is integrable, is then $f\circ \psi$...
Let$$\psi(x)=\left\{\begin{array}{cll}x \sin\Big(\dfrac{1}{x}\Big) & \text{if} & x\in (0,1],\, \\0 & \text{if} & x=0,\end{array}\right.$$and let $f:[-1,1]\rightarrow \mathbb{R}$ be...
View ArticleHow to show a function is differentiable n times.
If $f(x)$ is differentiable $n$ times, how to show $f(x^k)$ is also differentiable $n$ times. It is intuitively true, but chain rule does not work out very well, any advice?
View ArticleA continuous onto/surjective function from $[0, 1) \to \Bbb R$.
Does there exist a continuous onto/surjective function from $[0, 1) \to \Bbb R$?Finding difficult to site an example...
View ArticleDoes the deterministic sequence dominated by a random sequence converges to...
I am new to the concepts of different types of convergence of random sequences. Suppose $\{a_k\}_{k\in\mathbb{N}}$ is a deterministic sequence. Let $\{X_k(\omega)\}_{k\in\mathbb{N}}$ be a random...
View ArticleReal analysis book reference
What are the best books to self study real analysis? I am a physics masters student and am looking forward to study representation theory. I want to study the real analysis I need for studying...
View ArticleConvergence of Step Functions Generated by Uniformly Distributed Random Points
I have encountered a surprisingly complicated problem to solve and I'm looking for some help. It could be difficult because I don't have a background in probability and so don't know the appropriate...
View ArticleCalculate $\sum_{n=1}^\infty (1-\alpha) \alpha^nP^{*n}(x)$
$$\mbox{Let}\quadP'(x)=\sum_{j=1}^n a_j\frac{a^jx^{j-1}}{(j-1)!}e^{-ax}$$be the density function of a mixture of Erlangs and let $\alpha\in(0,1)$:Is is possible to determine an analytic expression for...
View ArticleApproximates $\sin{x}$
I have to find a neighborhood of $x=0$ such that $\sin{x}$ is approximated with an error less than $10^{-n}$.I have thought that from lagrange error I have:$$|\sin{x}-x|\leq...
View ArticleProving cluster points for a sequence [duplicate]
I have the sequence $a_n = ⌊\cos(\sqrt{n})⌋, n \in ℕ_0 $, and I have to determine the amount of cluster points. Of course, $\cos(x)$ oscillates between $-1$ and $1$, as $x$ varies, and therefore...
View ArticleHow to use local approximation spaces to build a global space via the...
I'm working to understand how the partition of unity method is used to build a global PUM space from local approximation spaces. Can someone please explain the mechanics of gluing the local spaces...
View Articlea Cauchy Schwarz application [duplicate]
Here an inequality that I feel that CS may prove it but I can't find the right way to use it :$\left (\sum_{i=1}^{n}a_{i}\right )^{2}+\left (\sum_{i=1}^{n}b_{i}\right )^{2}\leqslant \left...
View ArticleThis statement is true or false? [closed]
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. If $\lim\limits_{x\to\infty} |f(x)|=\infty$, then $\lim\limits_{x\to\infty} f(x)=\infty$ or $\lim\limits_{x\to\infty} f(x)=-\infty$. Is...
View ArticleDon't understand the solution
Question from chapter 1 of Putnam and BeyondShow that there does not exist a strictly increasing function $f :\mathbb N \to\mathbb N$ satisfying$f (2) = 3$ and $f (mn) = f (m) f (n)$ for all $m, n...
View ArticleConvergence to zero in L2 implies probability of being outside a bounded open...
Suppose I have a sequence of random variables $\{X_n\}_{n=1}^{\infty}$ taking values in $\mathbb{R}^{k\times k}$ and $S \subset \mathbb{R}^{k\times k}$ is a given bounded open set. If I know...
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