If $f\in X$, does $f^+$ and $f^-$ belong to $X$?
I ask a similar question to my previous one (which I do not know why has been removed).Let $X$ be an Hilbert space obtained through the closure of $C_c^{\infty}(\mathbb R, \mathbb R^n)$ with respect to...
View ArticleWhat's wrong with this solution to this Measure Theory problem?
Problem:Let ($X,\mathcal{A},\mu$) be a measure space and suppose $\mu$ is $\sigma$-finite. Suppose $f$ is integrable. Prove that given $\epsilon$ there exist $\delta$ such that$$\int_A |f(x)|\mu(dx)...
View ArticleFind $\lim_{n\to\infty}\frac{7^nn!}{n^n}$ [closed]
How can I calculate $$\lim_{n\to\infty}\frac{7^nn!}{n^n}\ ?$$I realize that $ n^n$ grows faster than each of the two factors in the numerator individually. So I thought the limit probably goes to...
View ArticleIs an arbitrary measurable function that vanish locally $\mu$-almost...
I came across this question when studying measure theory:Is an arbitrary measurable function that vanish locally $\mu$-almost everywhere integrable, with integral $0$?The motivation behind this...
View ArticleDecay at infinity of $L^2(\mathbb{R}^n)$ functions
I am trying to justify that a (normalized) solution $\phi$ in $L^2(\mathbb{R}^n)$ of:$-\Delta\phi+f(x)\phi=K\phi$, with $f(x)=0$ in $\Omega$, $f(x)=M$ in $\Omega^c$has to vanish outside $\Omega$ when...
View ArticleDoes a locally $\mu$-null set have measure 0?
A set $N$ is called locally $\mu$-null if for each set $A$ that belongs to $\mathscr{A}$ and satisfies $\mu(A)<+\infty$ the set $A\bigcap N$ is $\mu$-null. I have always been in trouble...
View ArticleHow to show that a subset of a normed space isn't dense?
Consider an arbitrary normed vector space $(X, \Vert \cdot \Vert_X)$ and let $Y \subset X$. We say that $Y$ is dense in $X$ if and only if$$ \forall x \in X, \epsilon > 0, \, \exists y \in Y : \Vert...
View ArticleComplex analysis or Real analysis book that have these special functions.
I saw an integral question that involved the digamma function, which I know nothing about, and I want to learn more about it and its properties and other functions like Polylogarithm function and...
View ArticleShortcut for computing the limit of an integral
Suppose we want to find $ lim_{x \to 0^+} \frac{1}{2x}\int_0^x ln(t)t^2dt $ without computing the integral. What I would do is this :We know that $ 0 \leq t^2 \leq x^2$ for all $t \in (0,x)$Then $...
View ArticleIf $g_1, g_2\in\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$ are equal locally...
BackgroundSuppose that $(X,\mathscr{A},\mu)$ is an arbitrary measure space, that $p$ satisfies $1\leq p<+\infty$, and that $q$ is defined by $\frac{1}{p}+\frac{1}{q}=1$. Let $g$ belong to...
View ArticleAn exercise from stein's fourier analysis
I'm trying to solve Exercise 20 of Chapter 5 of Fourier Analysis by Stein. The problem is as follows:Suppose $f$ is of moderate decrease and that its Fourier transform $\hat{f}$ is supported in...
View ArticleEvaluate $\int_0^1 \frac{\log(1+x)}{x}$
Evaluate the integral : $$\int_0^1 \frac{\log(1+x)}{x}dx$$It is an improper integral & I tried it by substituting $\log(1+x)=z$ . But it does not open any way to evaluate it.
View ArticleImplicit / inverse multivariate differentiation
I'm working on solving a pretty tricky optimization problem, where the constrain functions $g_i$ are not well behaved, so has been challenging for optimizers to work on directly. Instead of solving...
View ArticleConditions for differentiability of an infinite series of differentiable...
If $ (f_i(x))_{i=1}^\infty $ are each $\mathbb{R} \to \mathbb{R}$ differentiable functions on some interval, such that the series$$F(x):= \sum_{i=1}^n f_i(x)$$converges for all $x$ in the interval, is...
View ArticleDiscontinuous linear functional
I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional,...
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Range of $2-x\tanh^{-1}(x)$
To find the range of the function ( f(x) = 2 - x \tanh^{-1}(x) ) over the interval ( (0, 1) ), let's analyze its behavior by finding the limits and evaluating the function at the boundaries.Find the...
View ArticleUpper semicontinuity of sequence of Hausdorff measures
Let $(d_j)$ be a sequence of metrics on a compact space $X$ such that there exists $c,C>0$ with$$ cd_0(x,y) \leq d_j(x,y) \leq Cd_0(x,y) $$for all $x,y\in X$.Suppose also that $(d_j)$ converge...
View ArticleRoss Elementary Analysis Theorem 11.2 proof
The following is part of a proof for theorem 11.2 in Ross's Elementary Analysis.Let $(s_n)$ be a sequence and $t \in \mathbb{R}$. In the proof, we assume that\begin{equation}\{n \in \mathbb{N} : t -...
View ArticleApplying Ramanujan's master theorem to a general Taylor series
The Ramanujan master theorem states the following. If, for some analytic function $f$, there is a sufficiently nice function $\phi$ such...
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Lebesgue's Theorem for monotone functions
The proof is clear to me, but I don't understand how we can conclude the two things highlighted. I think it's something related to Vitali's covering, but I can't understand the reason. Would you be so...
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