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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Compactness of integral operators
came across this exercise I haven't been able to solve. I saw a very similar exercise (to prove an integral operator with a kernel is compact), but it is just different enough, I think, to warrant its ...
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Constructing an RKHS from a Kernel
I'm reading the book "High Dimensional Statistics" by Martin Wainwright just for fun (also as preparation of my PhD in computer science/Machine Learning). In particular, I'm currently ...
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On Schoen and Yau's proof of the positive mass theorem: extracting a minimal surface in the limit
I'm reading Schoen and Yau's 1979 paper on the Positive Mass theorem. I'm having trouble understanding the proof of how they extracted a minimal surface as the limit of solutions to the Plateau ...
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Do we have Arzela-Ascoli theorem on the space of all continuous stochastic process?
Let $\Omega$ be a polish space and $(\Omega,\mathcal{F},\mathbb{P})$ be a complete filtered probability space with the filtration generated by the standard Brownian motion. Let the space $U:=L^2\big(\...
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Arzela-Ascoli theorem on $C([a,b];\mathcal{H})$ with $\mathcal{H}$ a Hilbert space
To give some context: I need prove that a family $\mathcal{C}:=\{y: \|y(t)-y(s)\| \leq a(t) - a(s),\ \|y(t)\| \leq b(t)\}\subseteq C([0,T];\mathcal{H})$ is compact, for appropiated $a\in\operatorname{...
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Class of Lipschitz Functions on the unit d-dimensional ball
Let $\mathcal{F} = \{f:\mathcal{B}_d \to \mathbb{R}\;:\; \text{f is Lipschitz}\}$, where $\mathcal{B}_d = \{x \in \mathbb{R}^d\;:\: \|x\|_2 \leq 1\}$ is the unit ball in $d$ dimension. Is the class $\...
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Prove space of strict contractions is closed
I am given the following subset of $C([0,1])$ of continuous real-valued functions on $[0,1]$, satisfying
$$
\lvert f(x) - f(y) \rvert < \lvert x - y \rvert, \quad \int_0^1 f(x)^2 dx = 1
$$
(I am ...
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How to use Arzelà-Ascoli here?
The problem is
Suppose $\Omega\subset\mathbb{R}^{n}$ is a bounded open set. Consider the initial-boundary value problem $$\begin{cases}
\partial_{t}u(x,t)-\Delta u(x,t)=0 & \text{in}\ \Omega\...
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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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Family of continuous bounded functions has uniformly convergent subsequence
Let $\{f_n\} \subset C(-2,2)$ with $||f_n||_{L^\infty(-2,2)}\leq 1$. Assume for any $x,y\in (-1,1)$, there holds $|f_n(x)-f_n(y)|\leq |x-y|^\alpha $ for some $\alpha\in (0,1)$ whenever $|x-y|>\frac{...
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Continuity vs Uniform Continuity vs Pointwise Equicontinuity vs Uniform Equicontinuity
I recently started studying analysis and I'm (again) lost on some near concepts.
Intuitively, what is the difference between (i) Continuity, (ii) Uniform (iii) Continuity, Pointwise Equicontinuity and ...
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Countability of R in diagonalization argument of Arzela-Ascoli theorem proof
I have an intricate issue with the diagonalization argument used in the proof of Arzela-Ascoli theorem. It goes as follows:
So assume that $\scr F$ has these three properties [closed, bounded, ...
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Some problems in the application of Arzelà–Ascoli theorem
Let $\Omega$ be a bounded domain in $\mathbb{R}^d$ with a smooth boundary. Consider the sequence $\{u_n(\cdot,s)\} \subset L^2(0,T;L^2(\Omega))$ such that $\lVert u_n(\cdot,s) \rVert_{ L^2(\Omega)} $ ...
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Proving that $x_n(t)=\sin\left(n\frac{(t-n^2)}{n+1}\right)$ has subsequence that converges
I'm having some trouble with the following exercise:
Let $x_n:[a,b]\to\mathbb R$ be a sequence of functions given by$$x_n(t)=\sin\left(n\frac{(t-n^2)}{n+1}\right)$$ Prove that $x_n$ has a subsequence ...
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Equicontinuity and uniformly convergent subsequences
I have been trying to grasp equicontinuity lately and with it, the Arzela-Ascoli Theorem. It says that if ${f_k}$ is a sequence of functions on a compact interval that is uniformly bounded and ...
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Show that $\left\{x\in[0,1] \rightarrow F(x) = \int_{0}^{x} f(t)dt : f\in C([0,1])\text{ and }||f||_\infty \leq B\right\}$ is compact.
I am trying to show that $\mathscr{F}$ is compact, where $$\mathscr{F} = \left\{F(x) = \int_{0}^{x} f(t)dt\mid x \in [0,1],f\in C([0,1]) \text{ and } ||f||_\infty \leq B\right\}$$
To do so, I was ...
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Equivalent conditions of the Arzela-Ascoli theorem
I am taking functional analysis class and we stated the Arzela-Ascoli theorem in the following way:
Let $(K, d)$ be a compact metric space, $(Y, \| · \|_{Y} )$ a Banach space
and $\mathcal{F} \...
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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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A question related to a family of continuously differentiable functions
Consider $X=C^{1}([-1,1])$, the space of real valued $C^1$ functions defined on the interval $[-1,1]$. Define a norm on $X$ by $||f||:=\sup_{x\in[-1,1]}(|f(x)|+|f'(x)|)$. Consider the set $Y:=\{g:||g||...
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A question related to a family of continuous functions
Let $X:=\{g:[0,1]\rightarrow \mathbb{R}\:|\:g\in C^1,|g|\leq 10, |g^{'}|
\leq 10\}$. I am interested in showing that $X$ is a relatively compact subset of $C([0,1])=\{g:[0,1]\rightarrow \mathbb{R}\:|\:...
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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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Space of Lipschitz functions is finite dimensional
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: |...
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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
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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|$ ...
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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 ...
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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) = ...
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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 ...
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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 ...
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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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Solution verification of a proof of the Peano existence theorem, using Arzela-Ascoli
$\newcommand{\o}{\mathcal{O}}\newcommand{\d}{\,\mathrm{d}}$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 ...
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A version of Ascoli-Arzelà using modulus of continuity and nth entropy numbers
The classical Ascoli-Arzelà theorem could be stated as follows:
Let $K$ be a compact metric space and let $\mathcal{H}$ be a bounded subset of $C(K)$ - the space of continuous functions over $K$ with ...
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Sequence of contraction mapping and convergence of fixed point
Let $(𝑆,||_{\infty})$ be a metric space and $𝑇 : 𝑆→𝑆$ be a function mapping S into itself. $S$ is a space of bounded and Lipschitz continuous function.
For each $𝑛\inℕ$, $\tau_{n}\in T$ satisfies ...
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Variation of Ascoli-Arzelà theorem for $C^1$ functions
Let $\Omega \subset \mathbb{R}^n$ be an open set and let $(f_n)_n \subset C^1(\Omega)$ such that $\exists C > 0, \, \sup_{x \in \Omega} |f_n(x)| + \sup_{x \in \Omega} |Df_n(x)| \le C$ for all $n \...
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Arzelà–Ascoli $\implies$ Dini's theorem
If $K$ is compact Hausdorff then $f_n\in C_\mathbb{R}(K)$ with $f_{n+1}(x)\lt f_n(x) \quad
\forall x\in K$ and $f_n$ converges pointwise to a continuous limit $\implies$ $f_n$ converges uniformly
I ...
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Is the integral operator $I: L^1([0,1])\to L^1([0,1]), f\mapsto (x\mapsto \int_0^x f \,\mathrm d\lambda)$ compact?
Is the integral operator $I: L^1([0,1])\to L^1([0,1]), f\mapsto (x\mapsto \int_0^x f \,\mathrm d\lambda)$ compact?
For $I: L^p([0,1]) \to C([0,1])$ with $p\in (1,\infty]$ this can be shown quite ...
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Given a bounded sequence in $L^1([a,b])$, is $(t\mapsto \int_a^t f_n \,\mathrm d\lambda)_n$ equicontinuous?
We are given a bounded sequence $(f_n)_{n\in \mathbb N}$ in $L^1([a,b])$.
This means there is some $M>0$ such that for all $n\in\mathbb N$, $\int_{[a,b]} |f_n|\,\mathrm d\lambda \leq M$.
I wonder ...
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Arzelà-Ascoli for $C_b(0,1)$? Or more generally, why is that continuous functions "live most naturally" on compact spaces?
I’m wondering if there is a version of Arzelà-Ascoli for continuous functions on not-necessarily compact metric spaces $X$, i.e. a characterization of the compact subsets of $C_b(X)$ for $X$ not-...
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What's the role of domain compactness in the Ascoli-Arzela theorem?
The problem
From what I've read in the literature I get the idea that domain compactness is quite crucial in the theorem, and yet looking at the proof it seems to me as an unnecessary assumption. I ...
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Continuous extension in the proof of Arzela - Ascoli theorem
I'm reading a proof of Arzela - Ascoli theorem (complex analysis version) and I've come to the part where we want to extend a function that is defined on one set to the closure of that set. The ...
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Question about applying a combination of Arzela Ascoli and Schauder (also called Krasnoselskii)
Consider the equation
\begin{align}
3u(x) = x + u^2(x) + \int_0^1 | x - u(y) |^{1/2} \,dy \qquad (\ast)
\end{align}
Using Krasnoselskii's fixed point theorem, show that $(\ast)$ has a continuous ...
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If $(\varphi_n)$ is uniformly equicontinuous and $0\le f_n\le-3\ln\varphi_n$, show that $(f_n)$ is equicontinuous at $0$
Let $d\in\mathbb N$, $(\varphi_n)_{n\in\mathbb N}\subseteq C(\mathbb R^d)$ be uniformly equicontinuous with $\varphi_n(0)=1$ and $\varphi_n>0$ for all $n\in\mathbb N$ and $(f_n)_{n\in\mathbb N}\...
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How can we derive the Arzelà-Ascoli theorem for a $\sigma$-compact space from the ordinary Arzelà-Ascoli theorem?
Let $X$ be a compact topological space and $\mathcal F\subseteq C(X,\mathbb C)$.
By the ordinary Arzelà-Ascoli theorem, we know that
If $\mathcal F$ is relatively compact in the uniform topology, ...
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Pointwise convergence of $F_n(t) = \max_{y \in \mathbb R^N} \int_{B_t(y)} |u_n|^2 \ dx$
The following is a claim in some lecture notes I am reading:
Consider a bounded sequence $(u_n) \subset H^1(\mathbb R^N)$ with $\|u_n\|_{L^2} \to \lambda > 0$.
For $n \in \mathbb N$, the ...
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Set of functions on $C[0,1]$ with bounded derivative is compact
Just want to check if my proof works or not.
Question: Given $K > 0$, determine if the set
\begin{equation*}
Y = \{f \in C^1[0,1], f(0) = 0, \Vert f'\Vert_\infty\leq K\}
\end{equation*}
is ...
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