Theorem 6.17 in Baby Rudin, 3rd ed: $\int_a^b f \,d\alpha = \int_a^b f(x)...
Here is Theorem 6.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:Assume $\alpha$ increases monotonically and $\alpha^\prime \in \mathscr{R}$ on $[a, b]$. Let $f$ be a...
View ArticleShow that $\sum\limits_{k=1}^\infty\frac{(-1)^{k+1}}{k}x^{k-1}$ converges...
Show that $\sum\limits_{k=1}^\infty\frac{(-1)^{k+1}}{k}x^{k-1}$ converges uniformly on the closed interval $[0,1]$.It is clear that it converges pointwisely and we already know that a power series...
View ArticleQuestion About Dense Subspace of $L^p(X,\mathscr{A},\mu)$ - Proof of Theorem...
BackgroundI am self-studying Donald Cohn's Measure Theory second edition. I got stuck on a step of the proof of Theorem 4.5.1. Here is the theorem:Theorem$\quad$Let $(X,\mathscr{A},\mu)$ be a measure...
View ArticleLet $f:X\to Y$ be continuous and bijective. Prove/disprove that $f$ preserves...
I don't know how to start. Let $f:X\to Y$ be continuous and bijective. Prove or disprove that $f$ preserves interior points, that is $f(S^\circ) \subset f(S)^\circ$ for all $S\subset X$.Note: $A^\circ$...
View ArticleWhy is $f(x) = \frac{1}{x(|\ln x|+1)^2}\in\mathscr{L}^1$ but not...
So I am reading this post for an example of a function $f$ being integrable but $|f|^p$ is not integrable for $p>1$. But I couldn't see why$$f(x) = \frac{1}{x(|\ln x|+1)^2}$$is an example. Could...
View ArticleDoes $\lim_{x \rightarrow \infty} \frac {P(x)} {{Q}(\log x)} = \infty$ here?...
I am wondering whether the following statement is true or not: $\lim_{x \rightarrow \infty} \frac {P(x)} {{Q}(\log x)} = \infty$, where $P(x)$ and $Q(x)$ both obey the following constraints:They are of...
View ArticleIs $\sum_{n=1}^\infty\Gamma(n)/n^n$ demonstrably irrational?
The QuestionWhat do we know about $$\xi(z)=\frac{1}{\Gamma(z)}\sum_{n=1}^\infty \frac{\Gamma(n+z-1)}{n^n}= \int_0^1 \frac{dx}{(1+x\ln x)^z} $$Specifically, what can be said about $\xi$ regarding it's...
View ArticleImage of an increasing function with discontinuities on a dense countable set...
When discussing this question, we made interesting observations on increasing (i.e., nondecreasing) functions with discontinuities on rationals. Let $f: [0,1] \to [0,1]$ be an increasing...
View ArticleProof that $\|F_B\|\leq\|F\|$.
Background InformationLet $(X,\mathscr{A},\mu)$ be a measure space, let $p$ satisfy $1\leq p<+\infty$. Let $F$ be an arbitrary element of $(L^p(X,\mathscr{A},\mu))^*$, so $F$ is a continuous linear...
View ArticleIs $\lim\limits_{n\to\infty}\int_0^1\frac{n^kx}{(1+n^2x^2)^2}dx=0, \text{...
Is $\lim\limits_{n\to\infty}\int\limits_0^1\frac{n^kx}{(1+n^2x^2)^2}dx=0, \text{ where }k\in\{0,1,2\}$?My approach:We perform a substitution and...
View ArticleProving a Sequence Converges - Cauchy?
Let $a_n$ be q bounded sequence such that $$a_n-a_{n+1}\leq \frac{1}{2^n}$$ Let b_n be a sequence such that $$b_n=a_n-\frac{1}{2^{n-1}}$$Prove that $a_n$ converges.Prove that $b_n$ converges.I think...
View ArticleLet $f$ be continuous on $I=[a,b]$ such that $f(a)0$, $W= \{x\in I: f(x)
Let $f$ be continuous on $I=[a,b]$ such that $f(a)<0, f(b)>0, W= \{x\in I: f(x)<0\}$, $w=\sup W$ Prove that $f(w)=0$[I can see that this is an alternate proof for the Location of roots...
View ArticleThe set of Polynomials is not complete in sup-norm over [0,1]
Let $P$ be the set of all polynomials and consider the sup-norm on $[0,1]$ by $$||p||_\infty = \sup\{|p(x)| : x \in [0,1]\}.$$ I need to show $P$ is not complete in the sup-norm on the interval...
View ArticlePreserve adherent, boundary point [closed]
Problem: Let $f:X\to Y$ be continuous and bijective, for all $S\subset X$. Prove or disprove (a) $f$ preserve adherent point, ie $f(\overline{S})\subset \overline{f(S)}$; (b) $f$ preserve boundary...
View ArticleEquivalence of definitions for $L_\infty$ norm via essential range and...
I have the following two norms for mesurable complex valued functions $f \colon X\to \mathbb{C}$:$$\|f \|_1 := \inf \big\{ c \geq 0 \colon \mu (\{ x \in X \colon |f(x)| > c \}) = 0...
View ArticleLimit of $\left\lfloor\frac1x\right\rfloor$ as $x\to0$
I am trying to prove that the $$\lim_{x\to0}\left\lfloor\frac1x\right\rfloor$$ does not exist with the $\epsilon-\delta$ definition. I am struggling with how to structure the proof. I graphed the...
View ArticleImplicit function theorem and submanifolds in $\mathbb{R}^n$
I will use the following notations for tuples: $\mathbb R^{n+m} \ni (x, y) = (x_1, \dots, x_n, y_1, \dots, y_m)$. Furthermore, let $\mathbf f : \Omega \subset \mathbb{R}^{n + m} \rightarrow...
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Why does the definition of a one-to-one function contain an implication...
Reading Tao´s Analysis, I have seen the following definition of one-to-one functions:I am thinking about something rather simple here: Shouldn´t the implication be an equivalence instead? I thought...
View Article$-p\log p \prec H(p)p$: Any probability vector majorizes its associated...
I guess the following inequality$$\sum_{i=1}^n g (p_i) \ge \sum_{i=1}^n g \left (\frac{-p_i \log p_i}{H(\boldsymbol{p})} \right )$$holds for any continuous convex function $g$ and any probability...
View ArticleConterexample for Product of Absolutely Continuous Functions is Absolutely...
An function $f$ is absolutely continuous if for each $\varepsilon$ there is $\delta$ such that if $(a_i,b_i)_i$ are disjoint open intervals such that $\sum|a_i-b_i|<\delta$ then...
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