Representation of the Delta function, Riesz dual of evaluation at a point...
Consider the Sobolev Space$W^{1,2}([-1,1])$ (so one weak derivative, $L^2$-integrable)The linear map $ev_0:W^{1,2}([-1,1])\to \mathbb{R}$ defined by$$f\mapsto f(0)$$is well defined and continuous...
View ArticleShow that $\forall x,y \in (a,b):x
Let $a<b, (a,b \in \mathbb{R})$I want to show that there is an homeomorphism $f$ between $(a,b)$ and $\mathbb{R}$ such that $x < y \iff f(x) < f(y)$, $(x,y \in (a,b))$, but I don't want to use...
View ArticleContinuity without topology nor $\epsilon$-$\delta$
I was reading this question, and the related answers, which popped out in merit to my previous question about continuity while searching over here: $\lim_{x\to 0} f(x)$ where $0$ is isolated in the...
View ArticleContinuity in the topological sense, and singlets as open/closed sets
First of all I apologise if this question will sound stupid. I'm approaching to the study of topological spaces and real analysis in a deeper way, that is trying to fill holes and doubts and this one...
View ArticleAlmost sure convergence of conditional quantile
I am looking for similar results for point-wise convergence of quantile function, but with a different setting.Suppose $\{B_n\}$ is a series of random matrices, $B_n\in {R}^{m\times m}$ and $0\prec...
View ArticleTheorem Involving Taylor Expansion
Suppose $f:\mathbb{R}\to\mathbb{R}$ has n continuous derivatives. Show that for every $x_0\in\mathbb{R}$, there exist polynomials $P$& $Q$ of degree $n$ and an $\epsilon>0$ s.t....
View ArticleRelation of $\frac{f}{g}$ to $\frac{f'}{f}-\frac{g'}{g}$ in reverse order.
Suppose $f$ and $g$ have following properties: $f(0)=g(0)=0$, $f(1)=g(1)=1$, $f'(0)=g'(0)=0$, $f''(x)>0, f'''(x)>0, g''(x)>0, g'''(x)>0 \ \forall x\in(0,1)$.Then I want to show that...
View ArticleCan continuity be completly characterized by compact/connected sets?
We know that if $f : \mathbb{R} \to \mathbb{R}$ is a continuous function, then $f$ carries connected sets to connected sets and compact sets to compact sets. That is if $A \subset \mathbb{R}$ is...
View ArticleAverage of integral $xf(x)$ converges to zero [duplicate]
Let $f:[0,\infty)\to[0,\infty)$ be continuous and $\int_0^\infty f(x)dx<\infty$.Prove that$$\lim_{n\to\infty}\frac 1 n \int\limits_0^n xf(x)dx=0$$I have tried to approach this through integration by...
View ArticleHow can I naturally derive the Legendre transform formula? Why is the term...
A few months back I managed to find an article1 presenting the Legendre transform in a satisfactory way, in essence it beingThe Legendre transform $f^*$ of a (strictly) convex $C^2$ function $f$ is a...
View ArticleA monotone function has at most countably many discontinuities [duplicate]
Suppose that $f: S \rightarrow T$ is increasing, therefore $f(x-)$ and $f(x+)$ exist when $x \in S$. The fact to show is that $f$ has at most countably many jump points. A proof is given as follows on...
View ArticleLet $X$ and $Y$ be integrable, independent, and have an expectation of $0$....
This is an extremely difficult probability problem, and I have no idea how to handle this inequality.
View ArticleHow to showcase an inequality with a Dedekind cut of $\sqrt2$
I've been reading through Foundations of Analysis by Taylor, and I'm stumped on one part of the Dedekind cut section for formulating the reals. Right now, I'm on the section where it talks about a...
View ArticleFinding the factor $\Lambda(r)$?
I took the theta function$$\theta(\tau)= \sum_{n\in\Bbb Z} e^{\pi in^2\tau}$$and made the substitution $\tau=\frac{1}{\log t}.$ This gave me$$ \psi(t)=\sum_{n\in \Bbb Z} e^{\frac{\pi i n^2}{\log t}}....
View ArticleFolland Real Analysis Exercise 6.34
The question for referenceFolland 6.34 - If $f$ is absolutely continuous on $[\epsilon,1]$ for $0<\epsilon<1$and $\int_0^1 x|f'(x)|^p dx< \infty$, then $\lim_{x\to 0}f(x)$ exists(and is...
View ArticleConfusion on Dense set problem (Spivak, Chapter 8, Problem 6(a)), Guidance...
A set $A$ of real numbers is said to be dense if every open intervalcontains a point of $A$.My main confusion is with problem (c), I will post my proofs for (a) and (b) for background.(a) Prove that if...
View ArticleDoes such a weird convergent sequence exist?
Let $X$ be the Banach space of all $L^1$ functions $\mathbb{R}\rightarrow\mathbb{R}$. Take a convergent sequence $f_1,f_2,\ldots$ of functions from $X$. Let's call a sequence $g_1,g_2,\ldots$ of...
View ArticleEquivalent characterizations of continuous functions based on the graph of...
I had asked this question: Characterising Continuous functions some time back, and this question is more or less related to that question.Suppose we have a function $f: \mathbb{R} \to \mathbb{R}$ and...
View ArticleHow to prove that the improper integral is uniformly continuous?
Let $f$ be a uniformly continuous and bounded function on $\mathbb{R}$, and let $g$ be a continuous on $\mathbb{R}$ such that $\int_{-\infty}^\infty |g(x)| dx$ converges. Show that...
View ArticleProving the Equivalence of Two Definitions for a $C^1$ Boundary of an Open...
I want to prove the equivalence of the two definitions below:Definition 1: An open set $\Omega \subset \mathbb{R}^d$ has a $C^1$ boundary if, for each point $x \in \partial \Omega$, there exists a...
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