# 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).

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### Ascoli-Arzela theorem with incomplete codomain

Wikipedia (https://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem#Compact_metric_spaces_and_compact_Hausdorff_spaces) asserts the following (specialized from Hausdorff spaces to metric ...
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### Compactness of Differential Operator between Sobolev and $L^p$-spaces

I was wondering under which conditions the (weak) differential operator $D: W^{k,p}(\Omega)\rightarrow L^p(\Omega)$, $u \mapsto Du$ from a Sobolev space into the underlying $L^p$-space (on some open ...
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### Compactness of the immersion $C^1(\bar\Omega)\subseteq C^{\alpha}(\bar\Omega)$

Thanks to all who will answer me! (and sorry for my bad english...) I have to prove the compactness of the immersion $C^1(\bar\Omega)\subseteq C^{\alpha}(\bar\Omega)$ with (respectively) the usual ...
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### Compact embedding in $C^{k,2+\alpha}$ space

Could anyone please help me understand the compact embedding of $C^{k,2+\alpha}$ space? I am reading a paper about second order elliptic pde on $C^{k,2+\alpha}$ space. Here is the link of the paper ...
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### Arzelà-Ascoli and Compactness

Currently I'm reading baby Rudin and I've come across Arzelà-Ascoli Theorem (Theorem 7.25 in Rudin). I've read online that equicontinuity is exactly the additional property that we need to get ...
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### Prove that the integral operator $T:\mathcal{C}([0,1]) \to \mathcal{C}([0,1])$ is compact

Let $T:\mathcal{C}([0,1]) \to \mathcal{C}([0,1])$ be the operator given by $(Tf)(x) = \int \limits_{0}^{x} e^y \cdot f(y) dy, \: \forall f \in \mathcal{C}([0,1])$. where $\mathcal{C}([0,1])$ is ...
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### Some version of the theorem of Arzelà-Ascoli

I have a question concerning the theorem of Arzelà-Ascoli. Let $(f_n)_n:[0,T]\to \mathbb{R}$ be a family of functions so that $(f_n)_n$ is uniformly bounded on $[0,T]$ and $(\frac{d}{dt}f_n)_n$ is ...
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### $(X,d)$ separable implies uniformly equicontinuous subset of $C(X,\mathbb{R})$ is separable?

I'm trying to prove a certain class of subsets of $C(X)$ (made into a metric space, equipped with the sup-norm) are separable iff $X$ is (where $X$ is a metric space) and while one direction is easy, ...
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### Arzelà–Ascoli propagation theorem

Let $E$ be a metric space and $F$ be a Banach space, $A\subset E$ dense. Let $(f_{n})$ be a squence of continuous and bounden functions from $E$ to $F$ such that the restriction of $f_{n}$ to $A$ ...
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Let $(X,d)$ be a compact metric space. Let $V$ be a closed subspace of $C_{\mathbb{R}}(X)$ such that every $f\in V$ is Lipschitz. Show that $V$ is finite dimensional. Hint: Show that $A_n=\{f\in V: |... • 1,578 0 votes 0 answers 31 views ### Check the definition of "pointwise bounded under$d$" The above definition is from Munkre's topology in Section 45.I want to know whether the definition is correct?I think it contains a typo, in place of '$a$' it should be '$x$'... Please clarify this • 3,579 -2 votes 1 answer 151 views ### Uniform convergence via Arzela-Ascoli [closed] I want to show that$u_\epsilon = -\epsilon \log\left(\frac{ e^{\frac{x}{\epsilon}} + e^{-\frac{x}{\epsilon}}}{e^{\frac{1}{\epsilon}} + e^{-\frac{1}{\epsilon}}} \right)$converges uniformly to$1-|x|$... 1 vote 0 answers 59 views ### When is the compact-open topology on homomorphisms locally compact? Let$X$and$Y$be topological groups. The space$\mathrm{Cont}(X,Y)$of continuous functions will be given the compact-open topology. The subspace$\mathrm{ContHom}(X,Y)$of continuous homomorphisms ... • 4,401 2 votes 0 answers 220 views ### Set of functions with same Lipschitz constant attains a maximum Let$E$be the set of all functions$u : [0, 1] \to R$such that$u(0) = 0$and$u$satisfies a Lipschitz condition with Lipschitz constant$1$. Define φ : E → R according to the formula: $$\phi(u) = ... 0 votes 1 answer 77 views ### Proved that the given set is not closed in the function space \mathcal{C}([0,1]) The problem is actually taken from Davidson's Real analysis: Prove that the set S= \{ F:F(x) = \int_0^x f(t)dt, ||f|| \leq 1 ,\, f\in \mathcal{C}([0,1])\} is not closed. This means we should find ... 10 votes 2 answers 737 views ### When is the compact-open topology locally compact? Let X and Y be topological spaces, and consider the compact-open topology on C(X,Y), which is generated by open sets of the form$$\{\text{continuous }f\colon X\to Y:f(K)\subseteq U\}\text{ for ... • 4,401 3 votes 1 answer 62 views ### Extraction of pointwise convergenct subsequence using Arzela-Ascoli theorem Let$f_n:[a, b] \rightarrow \mathbb{R}$be a sequence of continuous functions which is uniformly bounded i.e.$||f_n||_{L^{\infty}} \leq M <\infty$and satisfies$f_n(a)=A$for all$n\in \mathbb{N}....
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$\newcommand{\o}{\mathcal{O}}\newcommand{\d}{\,\mathrm{d}}$I believe what I was required to show is a general version of the Peano existence theorem. Let $\o\subseteq\Bbb R\times\Bbb R^n$ be open and ...