Is exponentiation of isometries uniformly continuous in the uniform topology?
Let $(X,d)$ be a (bounded, complete, even polish) metric space, and let $G$ be the group of isometries of $(X,d)$ with the uniform metric $d_u$ defined by $d_u(f,g):=\sup_{x\in X}d(f(x),g(x))$ for all...
View ArticleEquivalence Relation Proof: Cauchy Sequences and Their Merged Sequence
I am working on a problem involving Cauchy sequences. The problem states:Two Cauchy sequences $\left(x_n\right)$ and $\left(y_n\right)$ are said to be equivalent if their merged sequence $\left(x_1,...
View ArticleSpivak Chapter 7, Problem-15
If $\phi$ is continuous and $\lim_{x \to \infty} \frac{\phi(x)}{x^n}=0=\lim_{x \to -\infty} \frac{\phi(x)}{x^n}$ then(a) Prove that if $n$ is odd then there is a number $x$ such that...
View ArticleProve that $g\in\mathscr{L}^q(X,\mathscr{A},\mu)$ when $\mu$ is...
Background InformationLet $(X,\mathscr{A},\mu)$ be a measure space, where $\mu$ is $\sigma$-finite. Let $p$ satisfy $1\leq p<+\infty$ and let $q$ be defined by $\frac{1}{p}+\frac{1}{q}=1$. Let $F$...
View ArticleApplication of Leibniz Rule [closed]
I want to show that if $f:[0,b]\times[0,b]\to\mathbb{R}$ is a continuous function, then$$\int_0^b\int_0^xf(x,y)dydx=\int_0^b\int_y^bf(x,y)dxdy.$$This exercise is in my analysis book in the Leibniz's...
View Articlehow does one prove that $|z_1 - z_2|^2 \geq \frac{1}{2} |z_1|^2 - |z_2|^2$
I am working on proving the following inequality for any complex numbers $z_1,z_2$$$|z_1 - z_2|^2 \geq \frac{1}{2} |z_1|^2 - |z_2|^2 $$I've attempted to expand both sides of the inequality and...
View ArticleLocal Boundedness and Lebesgue Integrability
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a Lebesgue measurable function. Suppose that for every $\epsilon > 0$, there exists a compact set $K \subset \mathbb{R}$ such that$$m(f^{-1}((-\infty,...
View ArticleChange of variable in the integral
Let $f:[0,b]\times[0,b]\rightarrow\mathbb{R}$ be continuous. Prove that $$\int_0^b dx\int_0^x f(x,y) \, dy=\int_0^b dy\int_y^b f(x,y) \, dx$$La idea es usar los teoremas:\ Regla de Leibniz, de...
View ArticleProving supremum through midpoints?
Proof:Let $S = (1,3)\cup(5,8)$Clearly $\forall x \in S,\;x<8$. Hence $8$ is an upper bound of $S$.To prove that 8 is the least upper bound suppose there exists an upper bound less than $8$.So...
View ArticleRadon-Nikodym derivative vs standard derivative. Multivariable case
For one dimensional case there is a nice connection of Radon-Nikodym derivative and "classical" derivative on real line. Is there some kind of analogy for higher dimensional cases?
View ArticleProving rigorously the supremum of a set
Suppose $\emptyset \neq A \subset \mathbb{R} $. Let $A = [\,0,2).\,\,$ Prove that $\sup A = 2$This is my attempt:$A$ is the half open interval $[\,0,2)$ and so all the $x_i \in A$ look like $0 \leq x_i...
View ArticleUnderstanding the role of supremum axiom when one wishes to prove supremum of...
To explain my point consider the set $\{ 1\} \subset \mathbb{R}$. I claim that the supremum of this set is $1$. If $x<1$, then it is not an upper bound of the set as, there is something in the set...
View ArticleA binary sequence such that sum of every 10 consecutive terms is divisible by...
Let $(a_n )_{n=1} ^{\infty} $ be a sequence of elements in $\{0, 1\}$such that for all positive integers $n,\Sigma_{i=n}^{ n+9}a_i$ is divisible by 3 . Then there exist a positive integer $k$ such that...
View ArticleConvergence of a certain bounded sequence
Let $(a_n)$ be bounded. Assume that $$a_{n+1} ≥ a_n - 2^{−n}$$ Then I need to show that $(a_n)$ is convergent.I can rewrite the above inequality as $$a_n - a_{n+1} \leq 2^{-n}$$ from which we get that...
View Article$\operatorname{argmin}(f_n)\to \operatorname{argmin}(f)$ if $f_n\to f$ uniformly
Let $X\subseteq \mathbb R$ be a compact interval and $f:X\to\mathbb R$ a continuous function with a unique minimizer such that$$x_0:=\operatorname{argmin}(f)\in X$$is well-defined. Now let $(f_n)_n$ be...
View ArticleIs the preimage of all sufficiently small neighborhoods of $\min(f)$...
Let $I$ be a compact interval and $f:I\to\mathbb R$ a continuous function with a unique minimizer. Is it true that the preimages of all sufficiently small neighborhoods of $\min(f)$ in $\mathbb R$ are...
View ArticleCompute an $\varepsilon$ smaller than a certain minimal distance of $n$ real...
Given $n$ positive real numbers $a_1,\ldots,a_n\in\mathbb R_{>0}$, we want to compute an $\varepsilon>0$ in polynomial time such that\begin{align}\sum_{i\in I_1} a_i \neq \sum_{i\in I_2} a_i...
View ArticleShow that if a subset of $C([0,1])$ is open relative to one norm it is also...
I am struggling coming up with a solution to the following question. Consider the two norms $||f||_1 = \int_0^1 |f(x)| \mathrm{d}x$ and $||f||_\infty = \sup_{x \in [0, 1]} |f(x)|$ on $C([0,1])$ (the...
View ArticleAn integral bound for two quadratic polynomials
Let us consider two quadratic polynomials dependingon two parameters$$F(x) = 1 + 2 p x + p q x^{2},$$$$G(x) = 1 + 2 q x + p q x^{2},$$where $x > 0$. Here $p > 0$, $q > 0$ are parameters.It is...
View ArticleProve that $x\sin x $ is continuous for all $x \in \mathbb R$ using the...
Prove that $x\sin x $ is continuous for all $x \in \mathbb R$, using the $\epsilon$-$\delta$ definition of continuity.I tried to do this but I get stuck at one point.My attempt:Let $x_0 \in \mathbb R$...
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