Prove that these two definitions of convex are equivalent
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. Given the following two definitions of convexity of $f$, prove that (i) implies (ii):(i) $\forall x, y \in \mathbb{R} : f(x)...
View ArticleA very complete proof on the separability of $L^p$.
I have to prove the following important result.Theorem. Let $(X,\mathcal{A},\mu)$ be a measure space such that:$(1)\;$ the measurable space $(X,\mathcal{A})$ is separable.$(2)\;$$\mu$ is sigma...
View Articlehi there please help with these real analysis 2 questions .
(a) Show that the intersection of an arbitrary collection of compactsets is compact.(b) Show that the union of a finite number of compact sets is compact.(c) Let (X; d) be a metric space and S is a...
View ArticleHow can I show whether the series $\sum\limits_{n=1}^\infty...
While revising for exams, I came across a question where at the end of it, we had to determine whether the below series was convergent or divergent:$$\sum_{n=1}^\infty \frac{(-1)^n}{n(2+(-1)^n)}...
View ArticleInfinite-dimensional normed vector spaces have continuous linear functional.
I'm reading Sheldon Axler's Measure theory book, and in a subsection dedicated to the Hahn-Banach Theorem, he states:In this subsection, we show that infinite-dimensional normed vectorspaces have...
View ArticleCan the inequality of integration be kept?
I'm considering two function $f,g\in L^{1}(\Omega)$ which satisfy\begin{align} \int_{\Omega}f dx\leq\int_{\Omega}g dx\end{align}Now, if I consider more a function $\phi\in C^{1}(\Omega)$ such that...
View ArticleClik here to view.
Some calculus questions regarding a reinforcement learning paper
I have a question to this paper from Reinforcement Learning, but I figured I ask it here because it involves mostly calculus. I am a bit confused because of some derivations.(1) In the following they...
View ArticleFind a function $\delta$ such that $(3x^3+x) \delta(x)^2 \rightarrow \infty$...
Let $f(x) := 3x^3+x$ and $g(x) := 9x^3+x$. Find a function $\delta$ such that $f(x) \cdot \delta(x)^2 \rightarrow \infty$ and $g(x) \cdot \delta(x)^3 \rightarrow 0$ for $x \rightarrow \infty$.I can see...
View ArticleIntegrate a sum of trig function under absolute value
Let $n \in \mathbb{N}$, I'm trying to compute an explicit formula for the following integral:$$\operatorname{I}\left(n\right) =...
View ArticleShowing an asymptotic $\sim$ relation for complicated terms
Let$f(x) = x + \frac{x^3}{3}$,$a(z) := x^3+x$,$b(z) := 3x^3+x$,$\delta(x) = \frac{1}{x^{4/3}}$.Is it true that$$f(xe^{i\delta}) \sim f(x) \exp\biggl(a(x)i\delta - b(x) \frac{\delta^2}{2} \biggr) \quad...
View ArticleProve that this sequence is eventually one
Let $ \ \mathbb{N} = \{ 0,1,2,3,4,...\} \, $, $ \ O = \{ n \in \mathbb{N} : n \text{ is odd} \} \ $ and $ \ T: O \to O \ $ be such that, for all $ \ n \in \mathbb{N} \, $,\begin{align*}T(8n+1) & =...
View ArticleOn locating inflection points
From what I have learnt, a point of inflection of a curve is, by definition, a point where the curve changes concavity.The Simple CaseThus, if, for a point, $c$, on a given function, $f(x)$, $f'(c) =...
View ArticleIntegrate and find closed form for $\int_0^\infty\frac{\sin x^n}{n\pi}dx$
I’m working on a practice exam for my analysis class, and I was asked to find a general form for$$\int_0^\infty\frac{\sin x^n}{n\pi}dx$$When I first looked at this, my mind instantly went to the...
View ArticleChoice of $q$ in Baby Rudin's Example 1.1
First, my apologies if this has already been asked/answered. I wasn't able to find this question via search.My question comes from Rudin's "Principles of Mathematical Analysis," or "Baby Rudin," Ch 1,...
View ArticleCorollary 5 in Royden-Fitzpatrick's Real Analysis: Convergence in Measure
Corollary 5: Let $\{f_n\}$ be a sequence of nonnegative integrable functions on $E$. Then$$\lim_{n \to \infty} \int_E f_n = 0 ~~~~~~(5)$$if and only if$$f_n \to 0 \mbox{ in measure on } E \mbox{ and }...
View ArticleHow to compute a function parameter such that an asymptotic estimate holds?
Let$f(x) = x + \frac{x^3}{3}$,$b(z) := 3x^3+x$,For some fixed $x \in \mathbb{R}_+$, can we find a $\theta_0$ with $\theta_0 < \lvert \theta \rvert < \pi$ such that$$f(x \cdot e^{i\theta}) =...
View ArticleProblem with the convergence region of the series
Can the region of convergence of a functional series defined on the real axis consist of a half-interval and a segment?Yes maybe! I understood this intuitively, but I couldn’t show it clearly. I...
View ArticleProve that a logarithmic function has maximum $0$
How do I prove that $$\log_{1/3} (|x-3|+1)$$ has maximum value $0$?Do I have to equate this log function and $0$ to find out if it equals $0$, or I need to solve this some other way?(Feel free to edit...
View ArticleQuestion about a branch of logarithm
I have a question about Daniel Fischer's answer (here)Why the function $g(w)$ is well-defined on $\mathbb{D} \setminus \{0\}$? I don't understand how $log$ function works here and how a branch of $log$...
View ArticleHaving trouble proving Taylor Expansion
In the book Introduction to Manifolds by Loring W. Tu, the author introduces a method for obtaining the Taylor expansion on "star-shaped" (i.e., any line segment connecting points in the set to a...
View Article