Explicit homeomorphism between $\mathbb{R}^n$ and $(0,1)^n$
I would like to prove that the map $f : \mathbb{R}^n\to (0,1)^n$ defined by$f(x)=(1_{(0,+\infty)}f_1(x_1)+1_{(-\infty,0])}f_2(x_1),...,1_{(0,+\infty)}f_1(x_n)+ 1_{(-\infty,0]}f_2(x_n))$ where...
View ArticleTopological game on $(0,1)$
I consider a « game » on a topological space with $2$ players. I will describe the game and tried to prove that one of the player has no winning strategy in the sense that the other player can always «...
View ArticleProving differentiability of a function defined by a series using convexity?
I am trying to solve the following problem:We are given $A=\{a_1,a_2,a_3,...\}$, a countable set of real numbers that is bounded. Let $$ f(x)=\sum_{n=1}^\infty{\frac{|x-a_n|}{10^n}}.$$Show that $f$ is...
View ArticleApproximate piecewise constant function with continuous function
I have a function $f(t)$ that is piecewise constant:$$f(t) = a_i \forall t\in[t_i,t_{i+1})$$with $n$ values $a_0, a_1, ..., a_{n-1}$, and $n+1$ values $t_0, t_1, ..., t_n$.I want to approximate this...
View ArticleAre Sobolev–Hölder functions continuous up to the boundary?
Let $U$ be an open subset of $\mathbb{R}^{n}$, let $k$ be a nonnegative integer, and let $W^{k,p}(U)$ ($1 \leq p < \infty$) be the Sobolev space consisting of all real-valued functions on $U$ whose...
View ArticleIs every $\alpha$-Hölder continuous function of bounded variation absolutely...
Let $f:[a,b] \to \mathbb{R}$ be $\alpha$-Hölder continuous function of bounded variation, does it follow that $f$ is absolutely continuous ? Here $\alpha \in (0,1)$ is fixed .Here are some sources :If...
View ArticleHow to prove that $f(x)=e^{x}$ is not a polynomial.
It is obvious that if we differentiate $f(x) = e^{x}$ with respect to x we will get again and again $e^{x}$. Can we conclude anything by considering the behavior at $\pm\alpha$
View ArticleConvergence/divergence of the series $\sum\frac{n}{(n+1)^{2}}$
The QuestionConclude the convergence/divergence of the series $\sum\frac{n}{(n+1)^{2}}$.My attemptI used derivative test to show that the function $\frac{x}{(x+1)^{2}}$ is a monotonically decreasing...
View ArticleClosed form for $\Gamma(a-x)$ where $a \in (0,1]$
I wonder if there is a closed form for $ \Gamma(a-x)$.The closed form is known when $a=1$ being the famous formula $\frac{\pi}{\Gamma(x)\sin(\pi x)}$ (and automatically the closed form exist for all...
View ArticleDerivative convergence rate from uniform convergence
Let $F_n: [-1, 1] \rightarrow \mathbb R$ be a sequence of infinitely differentiable functions that converges pointwise to $F: [0,1] \rightarrow \mathbb R$, which is also infinitely differentiable....
View ArticleWhy does the proof of Algebraic properties of norm in Rn space always start...
For instance simply to prove,||x|| = ||-x||Why do we need to start with||x||^2 = x.x = -x.-x = ||-x||^2And then take positive square root?Can't we just do,||x|| = sq rt (x.x) = sq rt {(-x) (-x)} =...
View ArticleShowing that $\sum_{i=1}^{\infty}2^{-k}\cos(kx)$ is continuous with a...
Here is a Real Analysis problem I have managed to develop a partial answer to and would be interested in the continuation:QuestionConsider the series $g(x)=\sum_{i=1}^{\infty}$$2^{-k}\cos(kx)$ and show...
View ArticleIs the inequality $abc\le \frac{1}{3}(a^3+b^3+c^3)$ true? [duplicate]
Let $a,b,c$ be real numbers in the interval $[0,2]$. Does the inequality $abc \le \frac{a^3+b^3+c^3}{3}$ hold? I'm reading a paper which claims this follows from the arithmetic mean-geometric mean...
View ArticleJacobian matrix of $f(\mathbf x) :=$ $\frac{\mathbf x}{\|\mathbf x\|} =$...
The following question comes from an introductory course on Multivariable Calculus. I have attempted the question but am seeking a solution verification as my attempt seems to be a little too...
View Article$-a=(-1)\cdot a,\forall a\in\mathbb{R}$ using axioms
I've been trying to prove$$-a=(-1)\cdot a$$for every $a\in\mathbb{R}$ using only axioms, however, every demonstration I found use one of these two properties:$$0\cdot a=0,\quad\forall...
View ArticleGiven a (nice) family of PDFs and a (nice) family of values, is there a...
Nice here is to be taken as smooth/continuous/differentiable/whatever. I'm going to just write out the integral here as I'm not as comfortable with EV notation.Say I have a (nice) function $p: M \times...
View ArticleLower bound function with first non-zero derivative
Let $f:[0,1]\to \Bbb{R}_+$ be real analytic. In other words, it has a convergent Taylor Series/Power series (but may have complex coefficients). Additionally, impose the following conditions on $f$:$1....
View ArticleChoice of a constant in Lieb and Loss' text on Analysis
I am currently reading Lieb-Loss' book on Analysis. In the proof of Theorem 1.9 (Brézis-Lieb Lemma), whose statement is not relevant here, they use the following statement:Let $p \in (0,\infty)$. For...
View ArticleThe Closure of a Subset of a Metric Space is the Union of the Subset and Its...
I'm trying to prove the following statement. I can prove it with words, but I'm trying to see how the logic checks out.Let $(\Omega ,d)$ be a metric space. Assume the following definitions of the set...
View ArticleA doubt in the proof of the spectrum of convolution operator
The following is a link showing the spectrum of convolution operator $A f(x)= \int_{-\pi}^{\pi} h(x-y) f(y) dy$$L^2( {-\pi},{\pi})->L^2( {-\pi},{\pi})$ is range of $h$ where $h$ is continuous...
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