Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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).

0
votes
0answers
22 views

Motivation for Arzela-Ascoli's theorem

I'm studying by myself Arzela-Ascoli's theorem and I'm reading this chapter of a lecture notes. Firstly, I would like to be clear that I know that the motivation of Arzela-Ascoli's theorem is ...
1
vote
1answer
24 views

If $F$ is a family of probability measures on $C(K,E)$ and $π_t$ is the evaluation map at $t∈K$, can we show that $(μ∘π_t^{-1})_{μ∈F}$ is tight?

Let $K$ be a compact metric space, $E$ be a separable metric space, $C(K,E)$ denote the space of continuous functions from $K$ to $E$ equipped with the supremum metric and $$\pi_t:C(K,E)\to E\;,\;\;\;...
0
votes
1answer
30 views

Equivalent conditions for relative compactness in the Skorohod space

Let $(E,d)$ be a complete separable metric space and $\mathcal X$ be a family of càdlàg functions $E\to[0,\infty)$. Consider the following claim: For all $t\in[0,\infty)\cap\mathbb Q$, there is a ...
5
votes
3answers
90 views

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'...
0
votes
1answer
19 views

Lemma related to proof of Montel's Theorem

I have seen the following lemma in my Complex Analysis class: Let $D \subset \mathbb{C}$ be open and connected, let $(f_n)_{n \in \mathbb{N}}$ be a sequence of functions holomorphic in $D$. Assume: $(...
1
vote
1answer
34 views

Uniformly convergent subsequence of a uniformly bounded family of functions

I am trying to solve the following problem: Suppose $\{f_n\}$ is a sequence of functions that are continuous and differentiable on $[a,b]$. Suppose that the sequence $\{f_n\}$ and $\{f_n'\}$ are ...
0
votes
0answers
24 views

Sequence of solutions to a continuous ordinary differential equation in $\mathbb R^n$ has a convergent subsequence

How can I prove that a sequence of solutions to a continuous ordinary differential equation in $\mathbb R^n$ has a subsequence that converge to a limit ? I was thinking of using Arzela Ascoli theorem....
1
vote
2answers
43 views

Existence of Solution for Initial value Problem $y'(t)=\cos(ty), t>0$ and $y(0)=\frac{1}{n}, n\in\mathbb{N}$ using Ascoli Arzela

Consider the cauchy problem (1)\begin{cases} y'(t)=\frac{1}{1+ty}, & t>0 \\ y(0)=1+\frac{1}{n} & n\in\mathbb{N} \end{cases} (2)\begin{cases} y'(t)=\cos(ty), &...
0
votes
1answer
33 views

Show that a sequence admits converging subsequence

I don't know how to solve the following exercise. I think I should use Ascoli-Arzelà's theorem, but I don't know how. Let $\{ u_n\}_n$ be a sequence of functions in $C^1[0,1]$ with $u_n(0)=0$ for ...
3
votes
1answer
64 views

Compactness of an integral operator from $L^2$ to $L^2$

I want to prove that The operator (linear and bounded) $T: L^2(0,1) \rightarrow L^2(0,1)$, defined by: $Tu(x)=\int_0^1\sin(x^2+y^2)u(y)dy$, is compact. Just by using theory, it's an Hilbert ...
3
votes
2answers
52 views

Compact operator by proving Ascoli-Arzelà

I need to prove that this operator satisfies Ascoli-Arzelà's hypothesis. $T: C^0[0,1] \rightarrow C^0[0,1] $, defined $Tu(x)=\int_0^x a(x,t) u(t)dt$, where $a(x,t)=C^0 ([0,1] \times [0,1])$. ...
2
votes
2answers
50 views

Show that this operator is not compact using Arzela-Ascoli

Let $T:C[0,1]\longrightarrow C[0,1]$ defined as $Tx(t) =tx(t)$. I need to prove that this operator is not compact using Arzela-Ascoli (using the Sup norm). I already prove that if X is a bounded ...
0
votes
0answers
23 views

The weaker boundedness implies uniformly bounded

Theorem (Arzelà-Ascoli). Eevery bounded equicontinuous sequence of functions in $C^{0}([a,b],\mathbb{R})$ has a uniformly convergent subsequence. The question asks to generalize the theorem with the ...
1
vote
0answers
44 views

Use of the diagonal argument in the proof of the Arzela Ascoli theorem

I'm trying understand the proof of the Arzela Ascoli theorem by this lecture notes, but I'm confuse about the step II of the proof, because the author said that this is a standard argument, but the ...
0
votes
1answer
38 views

Uniform limit points of a sequence of oscillating functions

The sequence $$f_n(x):=\sin(x+n), \qquad x\in [0, 2\pi],$$ is relatively compact in $C([0, 2\pi])$ by the theorem of Ascoli-Arzelà. This means that there exist sequences $n_k\in\mathbb N$ and ...
0
votes
1answer
32 views

An extension of a corollary of the Arzela-Ascoli theorem for smooth functions

I'm trying generalize this corollary for the case which the sequence of functions $\{ f_n \}_{n \in \mathbb{N}}$ are defined on a bounded domain (open and connected) $U \subset \mathbb{R}^m$ ($m \geq ...
1
vote
1answer
88 views

Proof of Arzela's Theorem

I am doing problem 3 from section 45 in Munkres. The problem is Prove Arzela's Theorem, which states: Let $X$ be compact: let $f_n \in \mathcal{C}(X,\mathbb{R}^k)$. If the collection $\{f_n\}$ is ...
0
votes
1answer
15 views

The union of a sequence of funtions and its convergent point

If $\{h_n\}_{n∈N} ⊂ C ([a, b])$ is $\| \cdot \|_{\infty}$ convergent to $h$, then $A ={h_n}∪{h}$ is $\| \cdot \|_∞$-compact, $\|\cdot \|_∞$-closed, $\| \cdot \|_∞$-bounded and uniformly equicontinuous....
-1
votes
1answer
16 views

union of finite bounded set and uniformly bounded set is bounded

Let $A, G \subset C ([a, b])$, $G = \{g_1, g_2, ..., g_m\}$ (finite set). Prove that if: i) $A || .. ||$ $\infty$-bounded then $A \cup G$ too. ii) $A$ equicontinuous in $x_o$ then $A \cup G$ also. ...
2
votes
1answer
36 views

Show that the sequence $\phi_{n}(t)=\int_{0}^{1}{\sin^{2}(t-ns)g(s)ds}$ has convergent subsequence.

Let $g\in\mathcal{C}([0,1])$ and $$\phi_{n}(t)=\int_{0}^{1}{\sin^{2}(t-ns)g(s)ds}$$ Show that the sequence $\{\phi_{n}\}_n$ has a uniformly convergent subsequence on $[0,1]$. I tried to use the ...
0
votes
0answers
41 views

Question Regarding Proof of Arzela Ascoli.

I have just proved the following: However my teacher provided me with the following: Note that $\Omega$ is open connected subset of $C$. This is what i don't get. I know that on each compact ...
3
votes
2answers
66 views

Arzelà–Ascoli theorem for the space $C_b^k(\overline{\Omega})$

Let $S \subseteq \mathbb{R}^m$ and $C_b^k(S)$, for $k \in \mathbb{N}$, the set of continuous functions from $S$ to $\mathbb{R}$ with bounded and continuous partial derivatives of any order $\leq k$. ...
2
votes
0answers
41 views

Apply Ascoli-Arzelá to show that a set is compact

Let $K: [0,1] \times [0,1] \to \mathbb{R}$ be a continuous function. Define the space $\mathcal{C} = \{f:[0,1] \to \mathbb{R} \mid f\;\text{is continuous in}\;[0,1]\}$ with norm $$\Vert f \Vert = \...
1
vote
1answer
19 views

Show that there is $f_{0} \in \overline{\mathcal{F}}$ \such that $J(f_{0}) = \min_{f \in \overline{\mathcal{F}}}J(f)$

Let $\mathcal{F} \subset C([0,1],\mathbb{R})$ be a family of functions such that: $f'(x)$ exist for every $x \in (0,1)$, $\forall f \in \mathcal{F}$; $\sup_{f \in \mathcal{F}}|f(0)| < \...
2
votes
1answer
58 views

Prove that a sequence contains a uniformly convergent subsequence if the derivatives are bounded by a function

For $n\in \mathbb Z_{>0}$ let $f_n:[0,1]\to R$ be continuous and differentiable on $(0,1]$ with $$|f_n'(x)|\le \frac{1+|\ln x|}{\sqrt x}$$ on $(0,1]$ and $$\left|\int_0^1f_n(x)\,dx\right|\le 10$$ ...
2
votes
1answer
47 views

Compactness of sublevels

Let $(X,d)$ be a metric space, complete and separable. Suppose that there exists a function $f:X \to [0,+\infty]$ with compact sublevels. Define the function $F:C([0,1],X)\to [0,+\infty]$ in this way $...
0
votes
1answer
42 views

$M$ equicontinuous and closed set of $C([a,b],\mathbb{R})$. There is $m \geq 0$ such that $|f(a)| \leq m$ $\forall f$. Prove that $M$ is compact

Problem. Let $\mathcal{M}$ an equicontinuous and closed set of $C([a,b],\mathbb{R})$. Suppose that there is a $m \geq 0$ such that $|f(a)| \leq m$ $\forall f \in \mathcal{M}$. Show that there are $f_{...
0
votes
1answer
78 views

Use of Arzela-Ascoli theorem to ensure that there is a subsequence of a sequence of functions that converges to an infinitely differentiable function.

I'm trying understand how to prove this result. I read the answer given on the post, but there are three points of the answer that I didn't understand: Why $f'_{n_{k_{i}}} \rightarrow f'?$ I know ...
3
votes
2answers
147 views

Show that $\{f_n\}$ has a uniformly convergent subsequence.

Let the function $f_n$ : $[0,1] \rightarrow [0,1]$ satisfy \begin{align*} \vert f_n(x)-f_n(y)\vert \leq \vert x-y \vert \textrm{ whenever } \vert x-y \vert \geq \frac{1}{n} \end{align*} Show that ...
0
votes
0answers
58 views

Abstract formulation of the Arzela Ascoli Theorem

In some paper I read, it says we may use the abstract version of the Arzela Ascoli theorem to conclude for some sequence $f_k: I \times \mathbb{R}^n \rightarrow \mathbb{R}^n$, $I$ some interval, that ...
2
votes
1answer
105 views

Applying Arzelá-Ascoli to a family of Hölder continous functions

Let $f_n:[a,b] \rightarrow R$ continous Hölder functions such that $$|f_n|_\alpha=\sup_{x\neq y} \frac{|f_n(x)-f_n(y)|}{|x-y|^{\alpha}}\leq M$$ for every $n\in N$. Prove that there exists a ...
2
votes
1answer
66 views

If $|f'_n(x)|\leq \frac{1}{\sqrt x}$ and $\int_0^1f_n(x)dx =0$, then exists a subsequence which converges uniformly [duplicate]

Let $f_n:[0,1]\rightarrow \mathbb{R}$ be continuous functions such that: $|f'_n(x)|\leq \frac{1}{\sqrt x}$ and $\int_0^1f_n(x)dx =0$ for every $n\in N$. Prove that exists a subsequence of $(f_n)$ ...
0
votes
3answers
54 views

How to check the compactness of following sets?

(1) Let $K \subset M_n(\mathbb{R})$ be defined by $$K = \{A \in M_n(\mathbb{R})\mid A = A^T, \ \operatorname{tr}(A) = 1, x^TAx \geq 0 \text{ for all } x \in \mathbb{R}\}$$ Then $K$ is compact. (2) ...
5
votes
2answers
206 views

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 $B\subset C(K)$...
0
votes
1answer
44 views

showing existence of uniform convergent subsequence of functions (Arzela-ascoli applying?)

Let $\{f_n\}$ be a sequence of real-valued $C^1$ functions on [0,1] such that for all n, $|f'_n(x)|\le 1/\sqrt x$ ($0<x\le 1$) and $\int_{0}^{1}f_n(x)dx=0$ prove that $\{f_n(x)\}$ has a ...
0
votes
3answers
85 views

Cantor diagonal process in Ascoli's theorem proof

The proof of Ascoli's theorem uses the Cantor diagonal process in the following manner: since $f_n$ is uniformly bounded, in particular $f_n(x_1)$ is bounded and thus, the sequence $f_n(x_1)$ contains ...
3
votes
0answers
78 views

Showing a set is compact using Arzelà–Ascoli theorem

Let $f(x)$ be a bounded function on $[a,b]$ show that the set of functions $F(x) = \int_{a}^{x} f(z) dz$ is compact but the set of functions $G(x) = \frac{d}{dx} f(x)$ is not compact. My trial Let ...
1
vote
1answer
70 views

Question about statements of Arzelà-Ascoli Theorem

I have some questions about the two versions of Arzelà-Ascoli Theorem given in Folland's book "Real Analysis" Version 1: Given a compact Hausdorff space $X$, let $F$ be an equicontinuous, ...
0
votes
1answer
15 views

Show that $\phi$ achieves its maximum on the set of $1-$ Lipschitz functions in $C[0, 1]$ passing through the origin

Let $E = \{u \in C[0,1]: |u(x)-u(y)|\leq |x-y|, x,y, \in [0,1], u(0)=0\}$ and let $\phi: E \to \mathbb{R}$ be define by $\phi(u)=\displaystyle\int_{0}^{1}(u(x)^2-u(x))dx.$ Show that $\phi$ achieves ...
1
vote
1answer
176 views

Arzela-Ascoli-type embedding: Is $H^1(0,T;X)$ compactly embedded in $C([0,T];X)$?

This is a variant of the question Compact Embedding of $W^{1,2}(0,T;ℝ^d)$ in $C(0,T;ℝ^d)$ where we had $X=\mathbb{R}^d$. Let now $X$ be some Banach space. Question: Is $H^1(0,T;X)$ compactly ...
3
votes
0answers
97 views

Compactness in $L^p$ and $L^\infty$

Let $(X,\Sigma,\mu)$ be a Borel probability space. Suppose we have a Banach space of functions on $X$ $(B,\|.\|)$ such that the unit ball of $B$ is compact in $L^p(X)$ for all $p>1$. Can we say ...
2
votes
0answers
59 views

Arzela-Ascoli argument on a partition

Suppose we have a bounded metric space $(X,d)$ and a countable measurable partition $Q$ of $X$ (with respect to some probability measure $\mu$) such that $\mu(q)>0$ for all $q\in Q$. We know that ...
0
votes
0answers
85 views

Definition of equicontinuity

In the literature I find several definitions for (uniform) equicontinuity of a family $M\in C[0,1]$. $M$ is called (uniformly) equicontinuous if ... (1) $\forall\varepsilon>0\;\exists\delta>0$ ...
1
vote
2answers
102 views

Characterization of Compactness in $\ell^\infty$

The metric space $\ell^\infty$ consists of all bounded real valued sequences with the metric is $ d(x,y) = \sup_{i \in \mathbb{N}} | \xi_i - \eta_i | $, where $x = (\xi_i), y = (\eta_i)$. I have been ...
3
votes
0answers
76 views

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 ...
1
vote
0answers
219 views

Proof of the Arzelà-Ascoli theorem

I have the following theorem: Let $f_n: X \to \Bbb{R}^k$ be a sequence of functions defined in a compact space $X$. If the collection $\{f_n\}_{n \in \Bbb{N}}$ is equicontinuous and if $\forall x \...
2
votes
1answer
206 views

Applications Arzela-Ascoli theorem

Dears, I am looking for some "nice" applications of the Arzelá-Ascoli theorem for the Banach space of continuous functions $x:[a,b]\longrightarrow X$, where $X$ is a Banach space. I known this ...
0
votes
1answer
95 views

Show that a sequence of functions has a uniformly convergent subsequence on (-1,1)

I'm studying for my Real Analysis qualifying exam and I'm a little unsure of one question. I'm given that $f_k$ is a sequence of continuous functions $(-1,1) \to \Bbb R$ such that $|f_k| \leq M$ for ...
3
votes
1answer
119 views

Show that closed subspace of differentiable functions is of finite dimension (using Arzela-Ascoli's, Riesz', and Banach's theorems)

Let $F\subseteq C^1([0,1],\mathbb{R})$ be a closed subspace of $C([0,1],\mathbb{R})$. Show that $F$ is of finite dimension. So, I considered the norm $\Vert f\Vert_1=\Vert f\Vert_\infty+\Vert f'\...
1
vote
1answer
178 views

Is an equicontinuous family of uniformly continuous functions necessarily uniformly equicontinuous?

Let $X,Y$ be metric spaces and $(f_n)_n$ a family of functions $X \rightarrow Y$. We say that $(f_n)_n$ is equicontinuous if $\forall x\in X \quad \forall \varepsilon >0 \quad \exists \delta >0 ...