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Continuous map obtained from smaller maps

Problem: Consider $X$, a metric space and $I=[0,1]$. Let $C(I,X)$ denote the set of continuous paths in $X$. Assume $A\subseteq C(I,Y)$ and that are given a continuous function $\delta:X\rightarrow...

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A lower bound for a quadratic form on $\mathbb{R}^2$

Let $a, c > 0$ and $b \in \mathbb{R}$. Show that for all $(x,y) \in \mathbb{R}^2$,\begin{equation*}a x^2 - 2bxy + cy^2 \ge \frac{ac - b^2}{2}\min(a^{-1}, c^{-1}) (x^2 + y^2).\end{equation*}This...

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If A is a Lebesgue Measurable set, find measure of reflection

Suppose we have a set $A \subset \Bbb{R^2}$ that is Lebesgue measurable.Now define $B$ = {$(x,-y) : (x,y) \in A$). We want to find the Lebesgue Measure of BIt’s seems to me that $B$ has the same...

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Proof Lebesgue-stieltjes eqals Riemann-Stieltjes integral in this case?

I am looking for a reference or a direct answer for a proof of this statement:Assume you have a random variable X. Define F(x) to be $P(X\le x)$,this function is bounded, monotone, non-deacreasing and...

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Some guidance needed, Spivak:Prove $\lim_{x \to \infty }...

Suppose $n=m$, as $x$ becomes larger and larger $f$ becomes closer to $1$ (Though I don't know how to show this rigorously). $$\forall \epsilon>0\exists N>0s.t(\forall...

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Riemann-Stieltjes integral over a degenerate interval.

On the Wikipedia article of Riemann-Stieltjes integration they require that $a<b$ if we integrate over the interval $[a,b]$. However, in Rudin's "Principles of mathematical" analysis they do not...

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Doubts on analytic functions

I have the following doubts about real analytic functions:If $f$ is analytic globally, can its radius of convergence be finite? In other words, if $f$ is analytic at every point, can the Taylor series...

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dist$(x,A)=0$ if and only if $x\in \overline{A}$ (closure of $A$)

Let $(X,d)$ be a metric space and non-empty $A\subset X$. The distance from $x$ to $A$ is defined as $$\mbox{dist}(x,A)=\inf\{d(x,a):a\in A\}$$I want to show that dist$(x,A)=0$ if and only if $x$ is in...

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Convergence in $L^p$ + boundedness of the gradient imply weak convergence in...

Say that $(u_n)_{n\geq 1}\subset W^{1,p}(\Omega)$ and $u\in W^{1,p}(\Omega)$ where $\Omega\subset\mathbb{R}^N$ is an open and bounded set and $\infty>p>1$. We know that:$$u_n\to u\ \text{in}\...

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A sequence such that $(a_{n+1} - a_n )\rightarrow \infty $ but...

Does there exist a sequence $( a_n )_{n\ge 1}$ of positive real numbers such that $\lim _{n \rightarrow \infty}(a _{n+1}-a_n) =\infty $ and$\lim _{n \rightarrow \infty}(\sqrt{a _{n+1}}-\sqrt{a_n}) =0 $...

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Weird result about seminorm in $\operatorname{BV}[a,b]$

Let $f\in \operatorname{BV}[a,b]$, where $\operatorname{BV}[a,b]$ stay for the set of functions $f:[a,b]\to \mathbb{R}$ of bounded variation. It follows that $f$ can be written as the difference of two...

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Monotonic increasing and restriction of a function in an interval

I have a function f : $[-1,1]\rightarrow R$. If I know that f is monotonic increasing in [-1,0] and in [0,1] then is monotonic increasing in all [-1,1]?I know that this isn't true for the function when...

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Restriction of a function and surjectivity

I have a function $f:[-1,1] \rightarrow \mathbb R$. If $f$ is surjective then restriction $f_{|[0,1]}:[0,1] \rightarrow \mathbb R $ is surjective?if I consider the function $f(x)=\tan({{\pi x} \over...

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Conceptual doubt about pullback and local coordinates

I'm studying differential forms in manifolds through the book: Differential Geometry and Topology With a View to Dynamical System, by Keith Burns.I have a conceptual question and I would like you to...

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A global implicit function theorem on a minimisation problem

Let $g(x,y)\in C^1(\mathbb{R}^n\times\mathbb{R}^n,\mathbb{R})$ such that the least eigenvalue of $\nabla_{yy} g(x,y)$ is always greater than $1$, $\nabla_{yy} g(x,y)$ is invertible and $\nabla_{yy}...

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Spivak's calculus on manifolds Partition of Unity

On Spivak "Calculus on Manifolds" p63 he provides a proof for the existence of a partition of unity but at some point in the proof for the compact case it looks like he already has the desired...

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$n$-th roots converge, but quotients do not [duplicate]

A known theorem states that for a sequence of positive real numbers $\{x_n\}$: $$\lim_{n \to \infty} \frac{x_{n+1}}{x_n} = L \implies \lim_{n \to \infty} \sqrt[n]{x_n} = L.$$I suspect that the reverse...

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Possible limits of subsequences in the ratio test

Let $0<y_{n}$ be a sequence such that $\frac{y_{n+1}}{y_{n}}\to C$for some $0<C<\infty.$ Let $x_{k}=y_{n_{k}}$ be a subsequence of$y_{n}$ such that $\frac{x_{k+1}}{x_{k}}\to B,$ for some...

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Evaluate $\int_0^{\pi/2}...

The following problem is proposed by Cornel Ioan Valean, that is, to prove that\begin{align}& \int_0^{\pi/2} \operatorname{Li}_{2}^{2}\left(\sin^{2}\left(x\right)\right){\rm d}x=\int_0^{\pi/2}...

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Solving an inequality problem involving $2^n$

Supposse that $0=<x_i=<1$ for $i = 1,2,..,n$ prove that $2^{n-1}(1+x_1x_2x_3...x_9) >= (1+x_1)(1+x_2)(1+x_3)..(1+x_n)$ equality holds when all $x_i$ equal to $1$My approach:$2 > (1+x_1)$...

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