Questions tagged [equicontinuity]

Let $(X, d_X)$ and $(Y, d_Y)$ be two metric spaces, and $\mathcal{F}$ a family of functions from $X$ to $Y$. The family $\mathcal{F}$ is equicontinuous at a point $x_0\in X$ if for every $\varepsilon > 0 $, there exists a $\delta > 0 $ such that $d_Y(ƒ(x_0),f(x) ) < \varepsilon$ for all $ƒ \in \mathcal{F}$ whenever $d_X (x_0, x) <\delta$.

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Sequence of 1-Lipschitz functions pointwise converge

I'm asked to prove the following: Let $(X, d_x)$ and $(Y, d_y)$ be two metric spaces. Let $D \subset X$ be dense. Show the following: If $f_1, f_2, \ldots$ is a sequence of $1$-Lipschitz functions so ...
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Sequence of contractions, Arzelà–Ascoli Theorem

Let $K$ be a compact subset of $\mathbb{R}^n$. Let $\{f_n\}$ be a sequence of contractions on $K$. We need to show that $\{f_n\}$ has a uniformly convergent subsequence. Initially, I struggled to ...
Nikola Golis's user avatar
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2 answers
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How to proove that $x \cos nx$ is equicontinuous?

How to proove that $x\cos(nx)$ is equicontinuous on $[0;1]$? I proved that is is equicontinuous on 0, however I cannot prove either it is equicontinuous on[0;1] or not. I also tried to do something ...
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Why uniform continuity does not imply equicontinuity?

Rudin's book "Principles of Mathematical Analysis" gives the following definition for equicontinuity: A family $\mathscr{F}$ of complex functions $f$ defined on a set $E$ in a metric space $...
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Show that the image of S under the map $ f \mapsto f * g $ is a compact set in $ C_0([-2, 2])$.

Given $\frac{1}{p} + \frac{1}{q} = 1 $, let $S = f \in L^p(\mathbb{R})$ $spt(f) \subset [-1,1]$, and $\|f\|_p \leq 1$ , and let $ g$ be a fixed but arbitrary function in $ L^1(\mathbb{R})$, with spt(...
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Uniform Integrability and relative compactness

I am trying to proof relative compactness in L2(0,1) for a specific set of functions $(\phi_n)_{n \in \mathbb{N}}$ with following properties: $\int_0^1 \phi(x) dx = 0 $ $||\phi_n^2||_{L1(0,1)} = 1 $ $...
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Family of equicontinuous function on a compact metric space $X$ to a metric space $Y$ is uniformly equicontinuous?

If $X$, $Y$ are metric spaces, $X$ is compact and $F\subseteq Y^X$ if $F$ is equicontinuous at any point $x$ i.e. for all $x\in X$ and $\varepsilon>0$ there is a $\delta>0$ such that when $d_X(x,...
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What approach to take to parameterise a C2, monotonic, piecewise polynomial?

I have a set of points (x, y) and need to monotonically interpolate between them. The resulting polynomial needs to be C2 continuous. Could you point me in the ...
Hiperfly's user avatar
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If $K(x,y)$ is continuous in $x$ and linear in $y$, is $\|K(x,\cdot)\|$ continuous?

Suppose that $H$ is a Hilbert space with norm $\|\cdot\|_H$, and $K : \mathbb R^n \times H \to \mathbb R$ is a function such that $K(x,y)$ is continuous in $x$ and a bounded linear operator in $y$. ...
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Equivalent metrics and equicontinuity [closed]

Let $(X,d)$ be a compact metric space, and $\varphi$ be an equicontinuous dynamical system ($\forall\, \varepsilon>0$, there exists $\delta>0$ such that $d(x,y)<\delta\Rightarrow d(\varphi^nx,...
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A uniformly bounded sequence of analytic functions converging on the boundary of the domain? [closed]

Suppose that we have a sequence of analytic functions $f_n:D(0,\rho_n)\to\mathbb C$ where $(\rho_n)_n$ is a decreasing sequence of real numbers $>1$ that converges to $1$. Assume furthermore that $|...
Leslie Molag's user avatar
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Uniform convergence, Pointwise Equicontinuity and Uniform Equicontinuity for functon.

I am trying to answer the following question: Define $f_n(s)=s^n$ where $f_n:[0,1]\rightarrow\mathbb{R}$. Is the family of functions $\{f_n\}$: Uniformly convergent Pointwise Equicontinuous ...
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Equicontinuity of $f_n(x)=n(e^{-\frac{1}{nx}}-1)$

The sequence of functions $f_n(x)=n(e^{-\frac{1}{nx}}-1)$ defined on $(0,\infty)$ have derivatives $f_n'(x)=\frac{e^{-\frac{1}{nx}}}{x^2}$. We then have $|f_n'(x)|<\frac{1}{x^2}$. On any interval $(...
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Equicontinuous and uniformly equicontinuous

Let $f:X\rightarrow Y$ be a continuous function between two metric space then the collection $\mathfrak{A}=\{f \}$ is equicontinuous if and only if... I found the question in a book named Topology of ...
Infinity's user avatar
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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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Hölder Condition and equicontinuity

My question is the following: If $|f_n(x)-f_n(y)|≤M|x-y|^\alpha$ for some fixed M and $\alpha>0$ and all x,y in a compact interval, show that ${f_n}$ is uniformly equicontinuous. As a criterion for ...
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Equicontinuity of a set of functions: contrexemple

Let $E$ be a Banach space, and $I=[0,1]\subseteq \mathbb{R}$. My goal is to construct a set of functions $X\subseteq\mathcal{C}(I,E)$ such that: $X$ is equicontinuous but not relatively weakly compact....
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Uniform continuous, equicontinuity, and Banach-Steinhaus

Banach-Steinhaus is a sufficient condition for a family of operators to be equicontinuous. Here, in Does uniform continuity on a compact subset imply equicontinuity? We have a function $f_n(x) = x^n$ ...
eggplant's user avatar
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If $\int_{-\infty}^{+\infty}|f_n(x)| + |f_n'(x)| + |f_n''(x)| \leq M$ then is $\{f_n\}$ equicontinuous and/or uniformly bounded?

I'm preparing for an exam and here's a practice question: Suppose $f_n$ is a sequence of differentiable functions on $\mathbb R$ such that $$\int_{-\infty}^{+\infty}|f_n(x)| + |f_n'(x)| + |f_n''(x)| \...
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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}\:|\:...
neophyte's user avatar
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3 answers
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Prove that $X$ is relatively compact in the space of continuous functions on $[-1,1]$

Let $X$ be the set of real-valued functions $g:\mathbb{R} \to\mathbb{R}$ of class $C^2$, with support in $[-1,1]$, and such that $\int_{-1}^1(g'')^2\le1$. I am interested in proving that the closure ...
neophyte's user avatar
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Equicontinuity is not an open condition

I am reading the book "Arithmetic Dynamics" from Joseph Silverman that has the following comment about the Fatou set: where the Fatou of a map means the Fatou set of its iterations. I could ...
Santiago Radi's user avatar
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Prove $(f_n) = (n(\exp(x/n)−1))$ is pointwise convergent, pointwise bounded and equicontinuous on $[0,1]$

For n ∈ $\mathbb{N}$, let $f_n : [−1, 1] \rightarrow \mathbb{R}, f_n(x) = n(\exp(x/n)−1)$. a) Show that the sequence of functions $(f_n)$ converges pointwise on $[−1, 1]$, and determine the limit ...
Ph_Ys321's user avatar
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Show that the sequence of functions converges uniformly.

I've had issues solving the following problem. The closest I have come is shown below. The problem is that in my "proof", I'm not using the fact that $K$ is compact, which makes me ...
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4 votes
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Determine which sequences are equicontinuous.

I'd appreciate if somebody could check if my proofs below are correct and also give me some hints on the equicontinuity of $(k_n)$ (part c)). $\mathbb{R}$ represents the real line with the standard ...
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An equicontinuous sequence of functions converging in $L^1$ converges pointwise a.e.

Let $f_n$ be a sequence of continuous functions defined on $\{x\in \mathbb{R}^d:|x|\leq 1\}$ such that $\|f_n\|_{1}\to 0$. Further, suppose that $f_n$ is equicontinuous. I need to show that $f_n\to0$ ...
omololo's user avatar
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Union of two equicontinuous families of functions is equicontinuous

Let $X$ be a metric space, $\mathcal{F}$ and $\mathcal{G}$ are families of real valued functions. Suppose that $\mathcal{F}$ and $\mathcal{G}$ are equicontinuous. I ever read that union of two ...
user136524's user avatar
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1 answer
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Arzela-Ascoli-like theorem with additional converging term

Let us say that I have a sequence of random $L^p$-valued functions $I^{\varepsilon} \in C([0,T], L^p(\Omega))$ such that $$\forall_{\varepsilon > 0}\ [[ I^{\epsilon}(t) - I^{\varepsilon}(s) ]]_p \...
defenestrator's user avatar
2 votes
1 answer
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Two equivalent versions of Arzelà–Ascoli theorem

Disclaimer: This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $(X, d)$ be a metric space, $C(X)$ ...
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Tightness vs Equi-integrability, Prokorov

I am a bit confused by this Theorem in the book 'Optimal Transport for Applied Mathematicians' of Santambrogio. This question concerns absolutely continuous (w.r.t Lebesgue) probability measures on $\...
non parratimo's user avatar
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let $\phi \subset C[0,1]$ be the set $\phi =\{f\in C^1[0,1]: \int^1_0 |f'(x)|^3dx\le 7\}$ prove that $\phi$ is equicontinous

let $\phi \subset C[0,1]$ be the set $\phi =\{f\in C^1[0,1]: \int^1_0 |f'(x)|^3dx\le 7\}$ prove that $\phi$ is equicontinous Def (Equicontunous ): A collection of functions on $\phi \subset C[0,1]$ ...
Inverse Problem's user avatar
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Check a family of functions is equicontinuous

Let $g_n: [0,1]\to \mathbb{R}$ for all $n\in\mathbb{N}$ where \begin{align*} g_n(x)= \begin{cases} n^2x,\quad 0\leq x\leq \frac{1}{n}\\ \frac{1}{x},\quad \frac{1}{n}< x\leq 1. \end{cases} \...
user136524's user avatar
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Proof of uniform convergence of converged equicontinuous sequence

Question: $\{f_n(x)\}$ is continuous on $[a,b]$ and has pointwise convergence on $[a,b]$ (to $f(x)$) . $\forall \epsilon>0,\exists\delta>0,\forall x,y\in[a,b](|x-y|<\delta),\forall n\geqslant ...
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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 ...
Thomas Browning's user avatar
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1 answer
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$K$ be compact metric space, $g$ and $h$ are continuous, prove the family of $f_u(x) = h(g(x)+g(u))$ is equicontinuous.

$K$ be compact metric space and $g: K \rightarrow R$ be continuous. $\forall u \in K$ let $f_u : K \rightarrow R$ be $f_u(x) = h(g(x)+g(u))$, where $h$ is any continuous function on $R$, show that the ...
mkom's user avatar
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Prove that the sequence of functions $f_n$ is equicontinuos, but with not uniformly convergent subsequence in $(0,1)$.

Question: Prove that the sequence of functions $f_n:(0,1)\to\mathbb{R}$ below is equicontinuous and simply converges to $f\equiv\dfrac{1}{2}$, but with not uniformly convergent subsequence in $(0,1)$. ...
turtleman's user avatar
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Is this proof that arguing equicontinuousness in a compact set is uniformly continuous valid?

This is a theorem that was discussed in class but I came up with another proof. The theorem is: Let $X$ and $Y$ be metric spaces. If $X$ is compact, and $h$ is a set of mappings from $X$ to $Y$ and is ...
Lythrum Sawyeraa's user avatar
10 votes
2 answers
591 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 ...
Thomas Browning's user avatar
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1 answer
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If $\{ f_n\}$ is equicontinuous on $X$ and bounded on dense set $D\subseteq X$, then $(f_n)$ is convergent on X.

Suppose $(X,d)$ complete metric spaces, $D\subseteq X$ dense in X, and $\mathcal{F}=\{f_n:X\to \mathbb{R}: n\in \mathbb{N}\}$ is equicontinuous on $X$. If $\mathcal{F}$ is bounded (above, below) on $D$...
user136524's user avatar
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1 answer
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Uniform convergence to zero implies equicontinuity

I am trying to understand the following explanation: prove/disprove: the functions $f_n(x)=\cos(x+n)+\ln\left(1+\frac{\sin^2(n^nx)}{\sqrt{n+2}}\right)$ are uniformly equicontinuous Particularly the ...
Addem's user avatar
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4 votes
1 answer
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Does a sequence of $C^1$ functions converge to a $C^1$ function if the derivatives are bounded and equicontinuous?

My question is: If $f_n\to f$ pointwise in $(0,1)$ where $f_n\in C^1$ (continuously differentiable), AND the family $\{f_n'\}$ is uniformly bounded and equicontinuous, then is $f\in C^1$ as well? Ie, ...
zz20s's user avatar
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1 vote
1 answer
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Uniformly equicontinuous sequence of functions that does not converge uniformly

In a past exam of a course I am taking this semester there is the following question for which I can't find any example by myself. The question is to find an uniformly equicontinuous sequence $(f_n)$ ...
azerty248's user avatar
2 votes
3 answers
83 views

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 ...
Michael's user avatar
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Equicontinuity + pointwise convergence implies uniform continuity

Let $f_n$ be an equicontinuous sequence of functions on a compact interval $D$ and suppose $f_n \to f$ pointwise. I wrote what I think is a solution, but I never used the assumption that $D$ was ...
user avatar
1 vote
1 answer
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$L^2$ Compactness with the Fourier Transform, Fréchet-Kolmogorov-Riesz theorem

I'm having difficulties proving the following. Let $G \subset L^2(\mathbb R^d)$ be bounded, meaning $\sup_{f \in G} \|f\|_2 <\infty$. Then the following two are equivalent: a) $lim_{R\to \infty}\...
Florian Ente's user avatar
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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 ...
blue's user avatar
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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, ...
0xbadf00d's user avatar
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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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