Questions tagged [arzela-ascoli]
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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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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: |... real-analysis metric-spaces lipschitz-functions arzela-ascoli- 878
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 general-topology solution-verification definition arzela-ascoli- 3,230
Uniform convergence via Arzela-Ascoli
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 $1-|x|$ ... uniform-convergence arzela-ascoli- 55
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 ... general-topology equicontinuity arzela-ascoli function-spaces- 3,824
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: $$ \phi(u) = ... real-analysis lipschitz-functions arzela-ascoli- 21
Proved that the given set is not closed in the function space $\mathcal{C}([0,1])$
The problem is actually taken from Davidson's Real analysis: Prove that the set $S= \{ F:F(x) = \int_0^x f(t)dt, ||f|| \leq 1 ,\, f\in \mathcal{C}([0,1])\}$ is not closed. This means we should find ... real-analysis metric-spaces uniform-convergence arzela-ascoli- 553
When is the compact-open topology locally compact?
Let $X$ and $Y$ be topological spaces, and consider the compact-open topology on $C(X,Y)$, which is generated by open sets of the form $$\{\text{continuous }f\colon X\to Y:f(K)\subseteq U\}\text{ for ... general-topology equicontinuity arzela-ascoli function-spaces- 3,824
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}.... analysis real-numbers sequence-of-function arzela-ascoli- 731
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 ... real-analysis metric-spaces solution-verification initial-value-problems arzela-ascoli- 9,395
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 ... functional-analysis approximation-theory arzela-ascoli- 3,852
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 ... functional-analysis fixed-point-theorems contraction-operator arzela-ascoli- 157
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 \... functional-analysis compactness arzela-ascoli- 381
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 ... real-analysis functional-analysis analysis arzela-ascoli- 415
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 ... functional-analysis compactness lp-spaces compact-operators arzela-ascoli- 309
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 ... functional-analysis lp-spaces equicontinuity arzela-ascoli- 309
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