Making Formal Substitution to a Complex Power Series
This problem asks me to:"Making the formal substitution $z-a=(z-z_0)+(z_0-a)$' in the power series $\sum_{i=0}^{\infty}A_n(z-a)^n$ and gathering like terms, obtain a series...
View ArticlePreparation for Folland/Royden/Graduate-Level Analysis
I’m trying to prepare for further study in Analysis and was wondering what advice you all would give. I have read (most of) Abbott’s Understanding Analysis and have started Rudin’s Principles of...
View ArticleIntuition behind taking limits in an inequality
I just came across a situation where $\mathbb{H}$ is a real Hilbert space and $\lbrace x_n \rbrace $ is a sequence in $\mathbb{H}$ that converges to some $\overline{x}$. Also, the sequence satisfies...
View ArticleProve that $\sup_y\int_{-\infty}^\infty\frac{dx}{(1+x^2)|x-y|}$ is obtained...
How to show that for a constant $0<\delta<1$, if $f$ is even, $f'(x)<0$ for $x>0$ and decays rapidly at infinity, then the supremum...
View ArticleThe condition "mass preserving maps" in the motivation of optimal transportation
I am reading Monge's formulation of the optimal transportation problem. It says that one wishes to find the a transport map $T: (X,\mu) \rightarrow (Y,\nu)$ that minimizes the transport cost $$\int_X...
View ArticleGraph of a continuous function has measure zero
I need help to solve the following problem:Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a continuous function. Prove that the graph$$G(f)=\{(x,f(x)):x\in\mathbb{R}^n\}$$has measure zero in...
View ArticleRudin's RCA Theorem 7.21
Theorem 7.21 states: If $f:[a,b]\to\mathbb{R}$ is differentiable at every point of $[a,b]$ and $f'\in L^1$ on $[a,b]$, then $f(x)-f(a)=\int_a^x f'(t)dt$ for all $x\in[a,b]$.From a very early theorem,...
View ArticleEquality of an integral of a measurable function
I have a question about an exercise in a book about measure theory/integration. In the first part I have to show that:$$\int_X f\,d\mu = \sup\left\{\sum_{k=1}^n \inf\{f(x):x∈...
View ArticleSum of limit inferior: $\liminf s_n + \liminf t_n \le \liminf (s_n+t_n)$
I'm seeking to prove the following statement:$$\lim \inf s_{n} + \lim \inf t_{n} \le \lim \inf (s_{n}+t_{n})$$provided that $s_{n}$ and $t_{n}$ are bounded.My solution so far:Given that $s_{n}$ and...
View Articlewhen do we say a metric space is quasi-invariant under a function?
A measure of a space that is equivalent to itself under "translations" of this space. More precisely: Let $(X,B)$be a measurable space (that is, a set $X$with a distinguished $ σ$-algebra $B$of subsets...
View ArticleThe existence of $f$ $k$-lipschitz in the subset $Y\subset \mathbb{R}$...
Here is the problem from a book for metric spaces I'm trying to solve:Let $f:Y\rightarrow\mathbb{R}$$k$-lipschitz in the subset $Y\subset \mathbb{R}$. Prove that there is a $k$-lipschitz function...
View ArticleDoes a particle undergoing perfectly elastic collisions within a rectangle...
The problem is: given a rectangle, a point mass is placed at any point on the sides of the rectangle. For all time t≥0, the point mass moves inside the rectangle, undergoing elastic collisions upon...
View ArticleProve the series $\sum_{n=1}^\infty (n(f(\frac{1}{n}) - f(-\frac{1}{n})) -...
Prove the series $\sum_{n=1}^\infty (n(f(\frac{1}{n}) - f(-\frac{1}{n})) - 2f'(0)) $ converges where $f$ is defined on $[-1,1]$ and $f''(x)$ is continuous. I already have a solution for this but I am...
View ArticleDoes rectifiability imply continuity?
I am trying to prove the following statement:Let $\mathscr{L}(x)$ denote the length of $f$ on $[a,x]$. If $f$ is rectifiable on $[a,x]$ for every $x\in[a,b)$ and $$\lim_{x\to b^-} \mathscr{L}(x)=L$$...
View ArticleIs f(x)=1/x is continuous and uniformly continuous on closed interval [0,1]?...
I'm not able to understand this question.please anyone explain..
View ArticleCalculate the normal cone of a set of increasing functions
Let $X:=\{f: [0,1] \rightarrow [0,1]\mid f \text{ continuous and increasing}\} \subset C[0,1]$.We can show that the set X is closed and thus compact, and also convex. (Roughly speaking, X can be...
View ArticleDistance of consecutive terms of a sequence goes to zero implies convergence...
Consider the vector space of real bounded functions $\Delta =$ {$f: \mathbb{R} \rightarrow \mathbb{R}: \text{ f is bounded}$}.Then the following is a norm on $\Delta$:$$\left\| f...
View ArticleIf f is increasing function on R and bounded on R then limit exists at...
I want to prove that, Let $f:\mathbb{R}\to\mathbb{R}$ is increasing and bounded, show that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to-\infty}f(x)$ both exists.And I don't know...
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An inequality involving $|(1/\zeta)^{(n)}(x)|$
I was plotting various derivatives of the function $(1/\zeta)(x)$ and I noticed that:$|(1/\zeta)^{(n)}(x)| \leq \frac{m(ln(2))^n}{2^x}$Holds for all $x \geq 2 , n\geq 1$ where x is a real number if...
View ArticleNorm on the space of rapidly decreasing and continuous functions
In P.Malliavin’s book "Integration and probability" a continuous function $f$ defined on $\mathbb{R}^n$ is said to be of rapidly decrease if for all integer $m$ we have that the mapping $(1+\lVert...
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