Study the equicontinuity of a family of functions

Question

Consider the family of functions $E=\{u_n:n \in \mathbb{N}\}$, where $u_n(x)=e^{-n||x||}$, $x \in \overline{B(0,1)} \subseteq \mathbb{R}^m$. I'm asked to study the equicontinuity of $E$ on the closed ball $\overline{B(0,1)}$.

This is my attempt:

First, we notice that the family is bounded with respect to the sup norm $||\cdot||_{\infty}$: $$||u_n||_{\infty}=\sup_{||x|| \le 1}e^{-n||x||} \le 1 ~~~ \forall n.$$ Then, by Ascoli-Arzelà theorem, if we prove that $E$ is not relatively compact in $(C(\overline{B(0,1)}),||\cdot||_{\infty})$, then we may conclude that $E$ is not equicontinuous. So, consider any subsequence $u_{n_k}$ of $E$. We have that $u_{n_k}(x)=e^{-n_k||x||}$ converges pointwise to the function $$u(x) =\begin{cases} 1 & x=0 \\ 0 & \text{otherwise} \end{cases}$$ which is not continuous, hence $u_{n_k}$ cannot converge uniformly on $\overline{B(0,1)}$ (because the uniform limit of a sequence of continuous functions must be a continuous function). Thus, $E$ is not equicontinuous on $\overline{B(0,1)}$.

Is my solution correct?

Answer 1

I think you can argue directly: let $\delta>0.$ Then, choose any $0\neq x\in B_{\delta}(0)$. Since $f_n(x)\to 0$, there is an integer $n$ such that such that $d(f_n(0),f_n(x_n))=d(1,f_n(x))>1/2$ so $\{f_n\}_n$ is not equicontinuous at $0$ and so cannot be equicontinuous.

Answer 2

It is correct.

Alternatively, we can explicitly show that your family is not equicontinuous at $0$. We claim that for $\varepsilon = \frac12$ there is no $\delta > 0$ such that for all $x \in \overline{B}(0,1)$ we have $$\|x\| < \delta \implies |1-u_n(x)| < \frac12$$ for all $n \in \Bbb{N}$.

Indeed, let $0 < \delta < 1$ be arbitrary and pick $n\in\Bbb{N}$ large enough such that $e^{-n\frac{\delta}2} \le \frac12$. Pick $x \in \overline{B}(0,1)$ such that $\|x \| = \frac\delta2$. We have $\|x\| < \delta$ but

$$e^{-n\|x\|} \le \frac12 \implies 1-e^{-n\|x\|} \ge \frac12$$ which proves the claim.