# equicontinuity

1. ### Sets of Functions, Functionals, and Equicontinuity

Real Analysis: Sets of Functions, Functionals, and Equicontinuity I originally asked my question on another forum and did not get an answer I was looking for. Here is the link to the question: real analysis - Sets of Functions, Functionals, and Equicontinuity - Mathematics Any help would be...
2. ### Equicontinuity implies Uniform Equicontinuity

Let A be a equicontinuous subset of C(X,F). Show that A is Uniform equicontinuous, ie: $$\forall \varepsilon \succ 0,\exists \delta \succ 0/{\left\| {f(x) - f({x_0})} \right\|_F} \prec \varepsilon ,\forall f \in A,\forall x,{x_0} \in X,d\left( {x,{x_0}} \right) \prec \delta$$...
3. ### Equicontinuity

Let X a metric compact space and Y a Banach Space. Let f and g $\in$ C(X,F) Prove that: 1) A+B is equicontinuous, and 2) A U B is equicontinuous I really dont know where to start....
4. ### Equicontinuity Question

Let f be a continuous function on \mathbb{R}. Let f_n = f(nt) for n \in \mathbb{N} be equicontinuous on [0,1]. I.e., \forall \epsilon >0, \exists \delta >0 such that if |x-y| < \delta, then |f_n(x) - f_n(y)| < \epsilon for all n. What can we conclude about f? All I am able to get is that...
5. ### Show uniform convergence implies equicontinuity and uniform boundedness

Hi guys! I was wondering if you could look at my proof and tell me if you think its correct/rigorous enough. I'm having a little difficulty with this. The question is: Let (f_n) be a sequence of functions in C([0,1]), with f_n uniformly converging to f in [0,1]. Show without using the...
6. ### Question regarding Equicontinuity and the open ball of functions

Hey - my hunch tells me that given an arbitrary open ball of functions (such as an arbitrary open ball centered at some f*) in the space of continuous functions defined on the compact interval [a,b] (ie, C0[a,b] sorry I don't know latex very well) with the standard sup norm, that this set is NOT...
7. ### Equicontinuity

Hi, I've been working on: Construct a bounded sequence of continuous functions f_n:[0,1] \rightarrow \mathbb{R} \\ s.t. \left| \left| f_n - f_m \right| \right| = sup \left| f_n(x) - f_m(x) \right| = 1, n \neq m, x \in [0,1] Can such a sequence be equicontinuous? So far I have the...
8. ### Uniform convergence and equicontinuity

This problem is giving me a headache. Suppose \{f_n\} is a sequence of functions defined on [a,b] which converges uniformly to f. Prove that \{f_n\} is equicontinuous. That is, show that whenever \epsilon >0, there is a \delta >0 such that if n is a positive integer and x,y \in [a,b] with...
9. ### need help with two (basic?) proofs involving equicontinuity

I'm stuck trying to wrap my head around two proofs involving equicontinuity. 1) If an equicontinuous sequence of functions (fn) converges pointwise to f on a set S, then f is uniformly continuous on S. 2) If a sequence of continuous functions (fn) converges uniformly on a compact set S, then...