Does application of epsilon delta definition for limits of a real function...
I believe it doesn't but it could be I am wrong.If we have a sequence $ \{\frac{1}{n}\}_n$ then the archimedian property plays a central role in proving the limit. For instance, let $\epsilon$ be an...
View Articlelet $V$ is inner product space.and let $B$ is closed subspace of $V.$ then is...
let $V$ is inner product space.and let $B$ is closed subspace of $V.$ then is it true $(B^\bot)^\bot=B$i know $B\subset(B^\bot)^\bot $.is the other way around is also true??
View ArticleBounding $N$ from below to simplfy $\epsilon-N$ proofs simply
When we do the epsilon delta proof for even the simplest of functions, it is a crucial step to restrict $\delta$ in some set of values before we give the actual value of delta example.I want to make an...
View ArticleWhat is the "procedure" mentioned here by Tao in Analysis I regarding...
I have 2 questions about a passage from Tao's Analysis I in which he defines recursive sequences. Questions have been asked about this passage here, here, here, and here; but I have different questions...
View ArticleAbsolute value of a Bochner measurable function is Bochner measurable?
Let $u:[a,b]\to H^1(\Omega)$ be a Bochner measurable function. We define:$$|u|:[a,b]\to H^1(\Omega),\ |u|(t)=|u(t,\cdot)|\in H^1(\Omega)$$Here we have used the well-known fact that the modulus of a...
View ArticleWhat IS the successor function without saying $S(n) = n + 1$?
I get really frustrated that all these real analysis books and online webpages say $S(n) = n + 1$ but then say addition is defined in terms off the relationships $a + 0 = a$ and $a + S(b) = S(a+b)$.I...
View Articleexistence of partial derivatives not closed under composition
Does there exist two functions, $f$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ and $g$ from $\mathbb{R}^2$ to $\mathbb{R}$ such that $f$ and $g$ has all partial derivatives at origin（but not...
View ArticleDistributivity of multiplication over infinite sum in extended non-negative real
Let $\mathbb{R}_{\ge 0}^{*} = [0, \infty]$ be the extended non-negative reals and multiplication be defined so that $0 \cdot \infty = \infty \cdot 0 = 0$. Does the following identity hold for any $k,...
View ArticleEvaluate...
I know some basic properties of the Dirac delta 'function', and I'm familiar with the sifting property. In any case, I'm stuck trying to evaluate this double...
View ArticleHow to find $\lim_{z\to\infty i}\sum_{n\in\mathbb{Z}}\frac{1}{n+z}$?
To be clear, I seek to find$$\lim_{t\to\infty}\sum_{n=\infty}^{\infty}\frac{1}{n+\alpha+it}$$where $\alpha\notin\mathbb{Z}$, $i=\sqrt{-1}$. I know this is just an expansion of $\cot$, but it looks like...
View ArticleQuestion About Absolute Continuity of A Finite Signed Measure (Proof of...
I am self-studying measure theory using Measure Theory by Donald Cohn. I got stuck on a step in his proof of Proposition 4.4.5. Here is the statement of the proposition:Proposition 4.4.5$\quad$Let...
View ArticleUnderstanding a proof that a noncompact topological space must have a subset...
I've encountered this claim and proof, and have a few questions about it.Let $(X, \tau)$ be a topological space, which is not compact. Thenthere is $C ⊆ X$, such that $C$ has no complete accumulation...
View ArticleOn the set of positive values of a non negative function
Let $f:\mathbb{R} \rightarrow [0 , \infty) $ be a function such that for any finite set $E\subset \mathbb{R}$ we have$\Sigma_{x\in E} f(x)\le 1 $Let $C_f=\{ x\in \mathbb{R} | f(x)>0\} \subset...
View ArticleInequality between limits inferior [duplicate]
Let $(a_n)$ be a non-zero real sequence with $\left (\frac{a_{n+1}}{a_n} \right)$ bounded. How might we prove that$$\liminf_{n\to\infty} \, \frac{|a_{n+1}|}{|a_n|} \leq \liminf_{n\to\infty} \,...
View ArticleMaximal lower bound of $|a_n|$, the coefficient of $x^n$ in a...
Inspired by this quesion. I'm curious about the maximal lower bound of $|a_n|$, where $n\geq 1$ and $a_n$ is the coefficient of $x^n$ in the symmetric-coefficient polynomial$$ P(x) =...
View ArticleDoes $\,x>0\,$ hint that $\,x\in\mathbb R\,$? [closed]
Does $x>0$ suggest that $x\in\mathbb R$?For numbers not in $\,\mathbb R\,$ (e.g. in $\mathbb C\setminus \mathbb R$), their sizes can't be compared. So can I omit "$\,x\in\mathbb R\,$" and just write...
View ArticleAccumulation point notion in the derivative definition
According to $[1]$ we have:Definition $\mathbf{13.1.3}$ (Derivative at $x_0$). Let $f：X\to \mathbb{R}$ be a real-valued function and$x_{0}\;\in\;X^{\prime}\;{\cap}\;X.$ The derivative of the...
View ArticleThe use of compactness in "compact plus closed equals closed" for metric spaces
I have seen some proofs/explanations for why "a compact set plus a closed set is a closed set". Some examples are: here, here, here, and here. However, I am confused about how we "leverage" compactness...
View ArticleWeird result about seminorm on $\operatorname{BV}[a,b]$
Let $f\in \operatorname{BV}[a,b]$, where $\operatorname{BV}[a,b]$ stay for the set of functions $f:[a,b]\to \mathbb{R}$ of bounded variation. It follows that $f$ can be written as the difference of two...
View ArticleEvaluating partial derivative by introducing a dummy variable.
Say I have a function $f(x)$, $x\in \mathbb{R}^n$ and I want to calculate:$\frac{\partial f}{\partial x}u$ (dot product). I saw some people do this: they define $x_{\alpha} = x+\alpha u$, where...
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