Measurability of sets used to define convergence in measure
In Folland's Real Analysis: Modern Techniques and Their Applications, the following definition is given for a sequence of functions converging in measure.We say that a sequence $\{f_n\}$ of measurable...
View ArticleAnalycity of $f*g$ with $f$ and $g$ smooth on $\mathbb{R}$ and analytic on...
Posted also on MO with a bountySuppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$analytic on $\mathbb{R}\backslash\{0\}$ but non-analytic...
View ArticleWhat is an "everyday" definition of a real number?
From what I understand, most mathematicians don't actually think of the rational numbers as equivalence classes of ordered pairs of integers—rather, that is how they are modelled in set theory. The...
View ArticleShow $0\epsilon\})\geq\delta_{p,\epsilon}$ for each $f\in E_p$.
Question: Let $(X,A,m)$ be a measure space such that $m(X)=1$. For each $1<p<\infty$ define the set $E_p=\{f\in L^1(m):\int |f|dm=1 \text{ and} \int |f|^pdm=2\}$. Show that for each...
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Question about equivalent statements of Picard Theorem
I want to prove following (Big Picard Theorem forms):Theorem.The followings are equivalent:a) If $f \in H(\mathbb{D}\setminus\{0\})$ and $f(\mathbb{D}\setminus\{0\}) \subset \mathbb{C} \setminus \{0,...
View ArticleProperties of the Bochner function space $L^{p \times q}([0,1] \times X)$
The function space $L^{p \times q}([0,1] \times X)$ is a variant of $L^p$-space which is a collection of functions whose norm $$\|f \|_{p \times q} := \| \| f(t, x) \|_{L^q(x:X)} \|_{L^p(t:[0,1])}$$ is...
View ArticleProving the Nested Interval Property using Axiom of Completeness
I'm self-studying real analysis using Abbott's text "Understanding Analysis." I'm trying to think out/prove as much on my own as I can, so I am working on proving the Nested Interval Property (Theorem...
View ArticleShowing the supremum squares to 2
In class, my real analysis teacher defined $\sqrt 2 = \sup \{x \in \mathbb{R} | x^2 < 2\}$ and left proving that this definition implies $\sqrt 2 ^2 = 2$ as an exercise. But I haven't been able to...
View ArticleBound of an integral function
Let $$f(r):=\int_{\mathbb{R}^d}\left|\int_{\mathbb{R}^d}e^{2\pi i \langle x,y\rangle}e^{-|y|^2+r^{1/2}|y|}dy\right|dx,$$for $r\geq 0$.Is it possible to uniformly bound $f$ on $\mathbb{R}_+$?...
View ArticleIntegrand of convergent integral - absolute continuity is necessary?
I think the following proposition is true (proof offered below)Proposition. Let function $f(x)$ be real-valued and nonnegative on $(a,\infty)$, and suppose that for $r > -1$ the...
View ArticleHow to find the roots of sin(x) using series theory
If we define $$\sin(x) = \sum_{n=0}^\infty \frac{(-1)^n \ x^{2n+1}}{(2n+1)!}$$How to find the roots of $\sin(x)$, i.e.$$\pi =4 \sum_{n=0}^\infty \frac{(-1)^n}{2n+1}$$satisfies $\sin (\pi)=0$
View ArticleGiven that $f(x)=\sum_{n=1}^{\infty} \frac{\sin(nx)}{n^2}$, express the...
Following my last question (Study the uniform convergence of $f (x) = \sum_{n=1}^{\infty} \frac{\sin(nx)}{n^2}$ in $\mathbb{R}$.), the second part of the problem goes as it follwos down below.Consider...
View ArticleProve that $\frac{(f^{\prime})^2}{f}$ is Riemann integrable on $\mathbb{R}$.
Let $f:\mathbb{R}\to\mathbb{R}, f\in\mathrm {C}^{\infty }(\mathbb {R})$ and $f>0$. It is known that the functions $f, f^\prime, f^{\prime\prime}, f^{\prime\prime\prime}$ are absolutely Riemann...
View ArticleProving the fundemental theorem of Calculus
I am currently trying to write a proof for the fundemental theorem of Calculus, mainly to practice writing proofs, and I have an attempt below. The problem is, I am unsure of how to interpret the...
View ArticleDirichlet problem for upper half plane
The Dirichlet problem I read is as follows:If $f$ is an integrable function, find a function $u$ such that for $x \in \mathbb{R}, y>0$ \begin{align}u_{xx} + u_{yy} & =0 \\\lim_{y \to 0^+} u(x,y)...
View ArticleReconciling metric and topological neighborhoods
Let $X$ be a metric space. Given a point in $x \in X$, an open neighborhood is more appropriately called an $\epsilon$-ball $N_\epsilon = \{p \in X : d(p, x) < \epsilon\}$, while a topological...
View Articleintersection of $(-1/n,1/n)$ as n$ \to \infty$ , is it empty?
What is the intersection of the set $(-1/n, 1/n)$ as $n \to \infty$?I got this idea of intersection from Intersection of open sets?, shouldnt the intersection be the empty set since a number cannot be...
View ArticleProof of Hadamard's Lemma in Duistermaat & Kolk's book
I am reading through Duistermaat and Kolk's book Multidimensional Real Analysis and I am having trouble with the intuition behind the proof of Lemma 2.2.7 (Hadamard's Lemma).The statement of the lemma...
View ArticleFunction whose gradient is of constant norm
Let $f:\mathbb R^n\rightarrow \mathbb R$ be a smooth function such that $\|\nabla f(x)\|=1$ for all $x\in \mathbb R^n$ and $f(0)=0$.I would like to prove that $f$ is linear.I first looked at the...
View ArticleA simple special case of Gronwall's inequality for Dini derivatives
Let $I=[t_0,t_1)\subset \Bbb R$ an interval and $a,b,c\ge0$ with $a>c$. Assume that $f\colon I\to\Bbb R$ is a continuous function with$$\tag{1}f(t)-f(s)\le \int_s^t\left( -af(r)+be^{-cr} \right) dr...
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