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Questions tagged [arzela-ascoli]

The Arzela-Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics. Use this tag alongside (real-analysis).

Consider a sequence of real-valued continuous functions $\{ f_n\}_{n\in \mathbb{N}}$ defined on a closed and bounded interval $[a, b]$ of the real line. If this sequence is uniformly bounded and equicontinuous, then there exists a subsequence $(f_{n_k})_k$ that converges uniformly. The converse is also true, in the sense that if every subsequence of $\{f_n\}$ itself has a uniformly convergent subsequence, then $\{f_n\}$ is uniformly bounded and equicontinuous.

229 questions
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When is the compact-open topology locally compact?

Let $X$ and $Y$ be topological spaces, and consider the compact-open topology on $C(X,Y)$, which is generated by open sets of the form $$\{\text{continuous }f\colon X\to Y:f(K)\subseteq U\}\text{ for compact }K\subseteq X\text{ and open }U\subseteq…
Thomas Browning
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Is there a version of the Arzelà–Ascoli theorem capturing $C([0,\infty))$?

I only know the Arzelà–Ascoli theorem for continuous functions on a compact topological space. However, in the context of characterizing weak convergence of probability measures on $C([0,\infty))$, I've seen that the following version is used…
0xbadf00d
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Proving an operator is compact.

I'm studying functional analysis and I ask this question mainly because I want a feedback on my proof and suggestions to formalize better my argument (if not already well formalized); any other hint to work on this kind of problem in a better way is…
Alberto Andrenucci
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Why is compactness required in Arzela Ascoli?

My question comes from the necessity of proving that one of the hypothesis in the Arzela Ascoli theorem fails in the example: $f_n(x)=\frac{1}{1+(x-n)^2}; x\in [0,\infty)$ I have tried proving it is not uniformly equicontinuous nor uniformly bounded…
PLanderos33
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Arzela-Ascoli Theorem on metric spaces

I've been looking for a proof of one particular direction of this theorem for metric spaces. I've looked online, but everyone seems to use different terminology/notation to state the theorem, so I'd like to ask for an outline of a proof specific to…
user308485
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Motivation for the Arzela-Ascoli's theorem

I'm studying by myself Arzela-Ascoli's theorem and I'm reading this chapter of lecture notes. Firstly, I would like to be clear that I know that the motivation of Arzela-Ascoli's theorem is to characterize when $E \subset \mathcal{C}(K,N)$ is a…
George
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Pointwise convergence for a sequence of convex functions implies uniform convergence

I'm trying to prove the following fact. Let be $(u_k)_{k\in\mathbb{N}}$ a sequence, where $u_k:\mathbb{R}^n\rightarrow (-\infty,+\infty]$ is convex. I suppose that $u_k$ converges pointwise to a function $u:\ \mathbb{R}^n\rightarrow…
GGG
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Flaw in Proof of Arzela-Ascoli in Carothers' Real Analysis?

I came across this overview of a talk given by a CalTech student which claims that the proof of Arzela-Ascoli given in Carothers' "Real Analysis" is incorrect. I've been trying to find the flaw for some time now but I don't see it. Maybe there isn't…
rosemary 2.0
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Arzela-Ascoli theorem exercise

The question: Define a metric space $C(K)=\left \{ f: K\rightarrow \mathbb{R} > \right\} $ , where $f$ is continuous function on $K$. Let $K\in \mathbb{R}$ be compact and let $B\subset C(K)$ be compact. Prove that $B$ is equicontinuousas…
Lee
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Direct method with integral constraint

Let $\Omega\subset\mathbb{R}^n$ be nonempty, open and bounded with $C^1$ boundary. Let $p\in[1,n)$. Let $g\in C(\mathbb{R})$ satisfy $$|g(y)|\leq C(1+|y|^q)$$ for some $C<\infty$ and some $q$ with $1\leq q
hannah2002
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Set of Hölder continuous functions is compact

Let $$H=\{x\in C[0,1]: \forall s,t\in [0,1]: |x(s)-x(t)|\leq b|s-t|^\alpha\}$$ where $C[0,1]$ are the continuous functions on $[0,1]$ with $x(0)=0$ and $\alpha\in (0,1/2)$. Take a sequence $(x_n)\in H$. Choosing…
user515599
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Arzelá-Ascoli Theorem precompact sets

Is the Arzelá-Ascoli Theorem true in a precompact subset of $\mathbb{R}^n$?. If $S \subset \mathbb{R}^n$ is precompact and we have a sequence $(f_n)$ of functions in $C(S)$ (Space of bounded and continuous functions $f: S \to \mathbb{R}$) which is…
Mike Van
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$f_n$ converges uniformly using Arzela Ascoli

I came across the following problem: Prove, with out using Arzela-Ascoli, that if $f_n:[0,1]\to\mathbb{R}$ is uniformly equicontinuous and $f_n\to f$ pointwise, then $f_n\to f$ uniformly. I know how to prove this without using Arzela-Ascoli, but I…
Tom Chalmer
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Under what metric spaces are pointwise and uniform convergence equivalent?

In exercise 1 of Charles Chapman Pugh's Real Mathematical Analysis, is a question of when pointwise and uniform convergence are 'equivalent' for metric spaces $M$ and $N$ and a sequence of functions $f_n: M \to N.$ Under my understanding, this…
joseph
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What is the difference between pointwise boundedness and boundedness?

In my textbook, the Arzela-Ascoli Theorem states that for a compact set $A$ in a metric space $M$ and $B\subset\mathcal{C}\left(A, N\right)$, where $N$ is another metric space, then $B$ is compact if and only if $B$ is closed, equicontinuous, and…
user281997
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