Is there a continuous nowhere differentiable function with...
In This question I askedDefine $g(x)= |x|$ for $|x|\in [-1,1]$ , $g(x+2)=g(x)$$$f(x)= \sum_{n \ge 1} \frac{3^n g\left(4^n x\right) }{4^n}$$ what is the $\sup \{\alpha\}$ such that$$\lim\limits_{h \to 0...
View Article$T:\ell^{\infty}\rightarrow L^{p}$ is continuous (norm-to-norm), is it...
Given norm-to-norm continuous map $T:\ell^{\infty}\rightarrow L^{p}$, $p\geq1$. Does norm-to-norm continuous imply weak*-to-weak continuous?I have learnt that: if $T$ is norm-norm continuous then it is...
View ArticleConvergence of the series $\sum_{n}\frac{3^{n}n!}{7 \cdot 10 \cdot 13 \cdots...
Question:Convergence of the series $\sum_{n}\frac{3^{n}n!}{7 \cdot 10 \cdot 13 \cdots (3n+4)}x^{n}$, $x >0$. I want to know its convergence at $x=1$.Context:I have found that the radius of the...
View ArticleShow that two subspaces $X$ and $Y$ of $\ell^1 (\mathbb{R})$ are such that...
Let $E = \left(\ell^1 (\mathbb{R}),\lVert \cdot\rVert_1\right)$ and consider the subspaces$$X = \left\{\left(x_n\right)_{n\in \mathbb{Z}_{>0}} \in E: x_{2n} = 0, \forall n\geq 1\right\},\hspace{1em}...
View ArticleHow can I evaluate the Gaussian Integral using power series?
It's a well known result that the Gaussian integral$$\int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$$evaluates to $\frac{\sqrt{\pi}}{2}$. This result can be obtained using double integrals with polar...
View ArticleQuestion about Lebesgue space for $0
We have defined the $L^p$ space for $p\geq1$.One thing I know is that for $0<p<1$ the norm will not satisfy the triangle inequality.$L^p(A)$={$f \colon A\to R$ (measurable) such that...
View ArticleFinding the limit of $a_{n} = \frac{n!}{\left(\frac{2}{7} +...
My task was to prove that the limit in the title exists and to calculate it. First I showed that the sequence is monotonically decreasing $\left(\frac{a_{n+1}}{a_{n}} = \frac{n+1}{\frac{2}{7} + n + 1}...
View ArticleUnable to come up with correct bounds for showing sequence convergence....
I was trying to prove the following (part (g) of the Theorem):Theorem 6.1.19 (Limit Laws). Let $(a_n)_{n=m}^{\infty}$ and $(b_n)_{n=m}^{\infty}$ be convergent sequences of real numbers, andlet $x,y$ be...
View ArticleUniform Convergence using Abel's test for a series based on convergence of a...
A problem from uniform convergence of series:$$\sum_{i=1}^\infty a_n$$ is convergent then show that $$\sum_{i=1}^\infty \frac {nx^n(1-x)}{1+x^n} a_n$$ and $$\sum_{i=1}^\infty \frac...
View ArticleProperty of Lebesgue Measure. Monotonicity of measure
I have been reading that a property of Lebesgue Measurable set $E$ is that, for each $\epsilon \lt 0$, there exist a family of open and disjoint intervals such that $$E \subseteq \cup_{n=1}^{\infty}...
View ArticleQuestion regarding the Kolmogorov-Riesz theorem on relatively compact subsets...
Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$.The following is from the book "Sobolev spaces" by...
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What is the probability that the absolute value of the roots of a polynomial...
Let $f(x) = 0$ be an equation of degree $n$. WLOG we can assume that the its coefficients are in $(-1,1)$. This is because we can divide each coefficient by the coefficient with the largest magnitude...
View ArticlePlancherel e(in)quality in more general spaces
Let $f \in L^2[0, 1]$, Plancherel's Equality says that $$ \|f\|^2 = \sum_{n = -\infty}^\infty |\hat{f}(n)|^2 = \sum_{n = -\infty}^\infty |\langle f, e^{2 \pi i n t}\rangle_{[0, 1]}|^2 .$$ Do a similar...
View ArticleProving $\lim_{n\to\infty} \left(1-2^{-\frac1n\log_2\frac n{\log...
Can someone help me prove that this limit equals $1/2$:$$\lim_{n\to\infty} \left(1-\left(\frac{1}{2}\right)^{\frac{1}{n}\log_2(\frac{n}{\log(n)})}\right)^{\frac{1}{\log_2(\frac{n}{\log(n)})}}$$I have...
View ArticleIf $A$ has strictly positive reach, does the set $\{ x \in A \colon...
Let $A \subseteq \mathbb{R}^n$ with $\text{reach}(A) > 0$ (see https://en.wikipedia.org/wiki/Reach_(mathematics) ).Define for any $\epsilon>0$, the "removal of $\epsilon$-thick...
View ArticleTangentspace of product of manifolds
I am currently trying to prove that for differentiable manifolds $M \subset \mathbb{R}^m$ and $N \subset \mathbb{R}^n$. It holds that:$T_{(x,y)} (M \times N) = T_x M \times T_y N$ for arbitrary $x \in...
View ArticleIs f(x) = x sin^2 (1/x) , f(0)=0 uniformly continuous on R? [closed]
Is $f(x) = x \sin^2\frac1x$ , $f(0)=0$ uniformly continuous on $\mathbb R$?I don't know whether $f(x)$ is uniformly continuous on $\mathbb R$ or not.Can I get some justification by the definition of...
View Article$A\subset\Bbb R^n\text{ open}, p\in\Bbb R^n-A$. Then $A\cup\{p\}$ is open...
Let $A\subset\Bbb R^n$ open and $p\in\Bbb R^n-A$, where $n>1$. Then $A\cup\{p\}$ is open if, and only if, $p$ is isolated on $\partial A$.Tried a few things but none of them worked as expected,...
View ArticleHow to show that $\int _0^{\infty}x^{4n+3}\exp(-x)\sin(x)dx=0$ for $n\in...
I want show that $\int _0^{\infty}x^{4n+3}\exp(-x)\sin(x)dx=0$ for $n\in \mathbb{N}$.How can I prove this?Do I need to use $\sin(x)=\Im(\exp(ix))$ and complex integral?I want to know how to prove it well.
View ArticleThe convergence of the Flint Hills series vs the convergence of...
The Flint Hills series, is the series $$\sum_{n=1}^\infty\frac{1}{n^3\sin^2(n)},$$ and it's an open problem as to whether the series converges. From the proof of Corollary 4 of this paper, it seems...
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