Prove that if $K \subset \mathbb{R}^{n}$ is compact, then every equicontinuous set $X \subset C(K;\mathbb{R}^{m})$ is uniformly equicontinuous.
I don't know any criteria to show that a given set is uniformely equicontinuous so, I tried to show this only by definition, but I could not. Would anyone have any suggestions?
Is it necessary to write down the definition of equicontinuous and uniformly equicontinuous set?
Edit. Using Daniel Fischer hint:
Proof. By hypothesis, for each $x \in K$, given $\epsilon > 0$, there is $\delta_{x} > 0$ such that if $x,y_{x} \in K$
$$\Vert x – y_{x} \Vert < \delta_{x} \Longrightarrow \Vert f(x) – f(y_{x}) \Vert < \epsilon,\quad \forall f \in X.$$
Note that $\bigcup_{x \in K} B_{\delta_{x}}(x)$ is an open cover of $K$ and this, has a Lebesgue Number, that is, there is $\delta > 0$ such that for every $x \in K$, exist an open $U$ of the cover such that $B_{\delta}(x) \subset U$. Thus, for every $x,y \in K$, if $\Vert x – y \Vert < \delta$ then $x,y \in U$ for some $U$ of the cover. Therefore
$$\Vert f(x) – f(y) \Vert \leq \Vert f(x) – f(y_{x}) \Vert + \Vert f(y_{x}) – f(y) \Vert < 2\epsilon,\quad \forall f \in X.$$
Then, $X$ is uniformly equicontinuous.
Is correct?
Best Answer
Consider the set $C(X,\mathbb{R}^m)$ of all continuous functions from $X$ to $\mathbb{R}^m$. On this set, consider the distance:$$d(f,g)=\begin{cases}\sup\left\{d\bigl(f(x),g(x)\bigr)\,\middle|\,x\in X\right\}&\text{ if }(\forall x\in X):d\bigl(f(x),g(x)\bigr)\leqslant1\\1&\text{ otherwise.}\end{cases}$$Now, consider the map$$\begin{array}{rccc}h\colon&K&\longrightarrow&C(X,\mathbb{R}^m)\\&k&\mapsto&\left(\begin{array}{ccc}X&\longrightarrow&\mathbb{R}^m\\f&\mapsto&f(k)\end{array}\right).\end{array}$$It is not hard to prove that $h$ is continuous if and only if $X$ is equicontinuous and that $h$ is uniformly continuous if and only if $X$ is uniformly equicontinuous. Now, use the compacity of $K$ and the fact that a continuous function from a compact metric space into a matric space is always uniformly continuous.