Auxiliar inequality for Rellich-Kondrachov theorem
To prove the Rellich-Kondrachov Theorem it is used the following statementIf $u\in W^{1,1}(\Omega)$, with $\Omega \subset \mathbb{R}^N$ open, bounded and s.t. $\partial \Omega$ is $C^1$, then...
View ArticleLet $\{a_n\}$ be a decreasing sequence of non-negative real numbers such that...
Let $\{a_n\}$ be a decreasing sequence of non-negative real numbers such that $\lim \inf (na_n)=0$ , then is it true that $\lim (na_n)=0$ ?
View ArticleDivergence of the harmonic series with logarithmic factor vs. convergent...
I'm trying to understand why the series $\sum_{n=2}^\infty \frac{1}{n \ln n}$ diverges. It's clear that for any $p > 1$, the series $\sum_{n=1}^\infty \frac{1}{n^p}$ (the p-series) converges....
View ArticleThe convergence of $\sum_{n=2}^{\infty} {1\over \ln(n!)}$
Check the convergence of $\sum_{n=2}^{\infty} {1\over \ln(n!)}$.I tried D'Alembert's test... Cauchy's test seems too intricate... I can't seem to understand what I should do here...
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Can I plug in numbers for x when proving limits?
At what point can I plug in values for x when proving limits? I know I can't have delta be a function of X. I also know delta should be epsilon divided by 9, but am I allowed to just start plugging in...
View Articledoubt on a limit of integral
i have $\lim_{n\to\infty}\int_{0}^{1}\frac{ne^x}{1+(nx)^2}dx$, which i computed by putting $nx=u$ and $\lim_{n\to\infty}\int_{0}^{n}\frac{e^\frac{u}{n}}{1+(u)^2}du$ by taking limit it gives me...
View ArticleWhat does $C^0([0,1])$ mean?
I am confused with what this denote. Does it mean the set of all continuous functions on $[0,1]$? I also find $C([0,1])$ in some books. Are they the same thing? Is $f_n=n\cos(x)$ in it?
View ArticleHow to prove that $\frac{\sin x}{x}$ is uniformly continuous at (0, 1)...
How can we prove that $\frac{\sin x}{x}$ is uniformly continuous at open interval $(0, 1)$ without using the mean value theorem?I've seen a lot of answers using MVT, but I cannot find how to do it...
View ArticleLooking for analysis books with in-depth treatment of Cantor Set and Cantor...
I'm a third-year mathematics undergraduate student currently taking a second course in Real Analysis.In the course, we were briefly introduced to the version of the Cantor Set being the set of all real...
View ArticleFinding a polynomial function for alternating m numbers of odds and even...
Context: The function $(-1)^n$ alternates between $1,-1$ because $n$ alternates between odd and even, I was trying to find a function $f(n)$ that will give $m$ positives then $m$ negatives and so on...
View ArticleWhy can we convert a power series of operators to a function, invert the...
I apologize as I am still somewhat unfamiliar with infinite series and knowing when and how we are allowed to use them, and especially with working with operators in this way. I imagine that I will be...
View ArticleA version of Helly's theorem without convexity but connectivity
It's more than 100 years since Helly's celebrated theorem in discrete geometry was published by him, and yet ghosts still remain. The theorem in it's infinitary glory statesLet $\{X_j\}_{j\in J}$ be a...
View ArticleComputing the integral of the sine integral remainder function
Define $\mathrm{si}(x) := \displaystyle\int_x^\infty \frac{\sin(t)}{t}\mathrm{d}t $ for all $x>0$. I have showed, by integration by parts, that this function has convergent integral over...
View ArticleComplex measure on $\mathbb{C}$
When people talk about complex measure on $\mathbb{C}$, is there a conventional one?For example, the theorem below is from Rudin's Real and Complex Analysis:Suppose $\mu$ is a complex measure on a...
View ArticleProving a lower bound for this double integral
Let $f:\mathbb R^n\to\mathbb R$ be a smooth function. Let $k>0$ and consider the cutoff function $T_k:\mathbb R\to\mathbb R$ defined for any $t\in\mathbb R$ as$$T_k(t)=\begin{cases}t-k &...
View ArticleSolution verification - Exercise 3.4.1 Terence Tao Analysis 1
Let $f : X → Y$ be a bijective function, and let $f^{−1}: Y → X$ beits inverse. Let $V$ be any subset of $Y$ . Prove that the forwardimage of $V$ under $f^{−1}$ is the same set as the inverse image...
View ArticleProof that the volume function is $\sigma$-additive
A $\textbf{box}$$Q\subset \mathbb{R}^n$ is defined as $Q:= (a_1,b_1) \times \ldots \times (a_n,b_n)$ and the same with closed or half opened intervals where $a_i < b_i \in \mathbb{R}$ and in case of...
View ArticleCan I swap the order of a limit and an antiderivative?
I'm trying to solve a math problem, and I do not know if one step is possible. Is this step valid?$\int \lim_{u \to x} f (u) dx = \lim_{u \to x} \left( \int f (u) du \right)$In case it's relevant, I...
View ArticleIs there a sequence such that $a_n\to0$, $na_n\to\infty$ and...
Is there a positive decreasing sequence $(a_n)_{n\ge 0}$ such that${\it i.}$ $\lim\limits_{n\to\infty} a_n=0$.${\it ii.}$ $\lim\limits_{n\to\infty} na_n=\infty$.${\it iii.}$ there is a real $\ell$ such...
View ArticleEquivalent definitions of affine function: Does $f(\gamma x...
For a function $f\colon\mathbb{R}^n\to\mathbb{R}$ the following two are equivalent\begin{align*}&\text{(i) there exist $a\in\mathbb{R}^n$, $b\in\mathbb{R}$ so that...
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