# Family of equicontinuous functions on compact set

Homework Statement .

Let ##X## be a compact metric space. Prove that if ##\mathcal F \subset X## is a family of equicontinuous functions ##f:X \to Y \implies \mathcal F## is uniformly equicontinuous.

The attempt at a solution.

What I want to prove is that given ##\epsilon>0## there exists ##\delta>0##: if ##d_X(x,y)<\delta \implies \forall f \in \mathcal F d_Y(f(x),f(y))<\epsilon##. I know that for an arbitrary ##x_0 \in X## and a given ##\epsilon## I can find ##\delta##. I also know that ##X## is compact. I think I should write ##X## as a union of open covers involving something with the ##\delta## that works for each point ##x_0##, then extract a finite subcover (I would have finite ##\delta##'s) and take the minimum of those deltas. I got stuck trying to find the proper union of open covers.

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pasmith
Homework Helper
Homework Statement .

Let ##X## be a compact metric space. Prove that if ##\mathcal F \subset X## is a family of equicontinuous functions ##f:X \to Y \implies \mathcal F## is uniformly equicontinuous.

The attempt at a solution.

What I want to prove is that given ##\epsilon>0## there exists ##\delta>0##: if ##d_X(x,y)<\delta \implies \forall f \in \mathcal F d_Y(f(x),f(y))<\epsilon##. I know that for an arbitrary ##x_0 \in X## and a given ##\epsilon## I can find ##\delta##. I also know that ##X## is compact. I think I should write ##X## as a union of open covers involving something with the ##\delta## that works for each point ##x_0##, then extract a finite subcover (I would have finite ##\delta##'s) and take the minimum of those deltas. I got stuck trying to find the proper union of open covers.

Let $x \in X$ and $y \in X$ be such that $d_X(x,y) < \delta$. You want to show $d_Y(f(x),f(y)) < \epsilon$. You have a finite subcover by open balls so there must exist some $z \in X$ such that $x \in B(z,r(z))$ for some ball $B(z,r(z))$ in the subcover, and the triangle inequality will then give you a bound for $d_X(y,z)$. You can then, by appropriate choice of $\delta$ and $r(z)$, ensure that you have bounds on $d_Y(f(x),f(z))$ and $d_Y(f(y),f(z))$, and a further application of the triangle inequality will give a bound on $d_Y(f(x),f(y))$.

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Let $x \in X$ and $y \in X$ be such that $d_X(x,y) < \delta$. You want to show $d_Y(f(x),f(y)) < \epsilon$. You have a finite subcover by open balls so there must exist some $z \in X$ such that $x \in B(z,r(z))$ for some ball $B(z,r(z))$ in the subcover, and the triangle inequality will then give you a bound for $d_X(y,z)$. You can then, by appropriate choice of $\delta$ and $r(z)$, ensure that you have bounds on $d_Y(f(x),f(z))$ and $d_Y(f(y),f(z))$, and a further application of the triangle inequality will give a bound on $d_Y(f(x),f(y))$.
Let ##\epsilon>0##, I know that for each ##x \in X##, there exists ##\delta_x##: if ##y \in B_X(x,\delta_x) \implies \forall f \in \mathcal F d_Y(f(x),f(y))<\frac {\epsilon} {2}##. If ##y,z \in B(x,\delta_x) \implies d_Y(f(y),f(z)<\epsilon##. I can write ##X## as ##X=\bigcup_{x \in X} B_X(x,\delta_x)##, which is an open cover of ##X##. By hypothesis, there exists ##\delta>0## a Lebesgue number for this open cover. This means that for any ##y,z## such that ##d(y,z)<\delta \implies y,z \in B_X(x,\delta_x)## for some ##x \in X \implies d_Y(f(y),f(z))<\epsilon##, this proves ##\mathcal F## is uniformly equicontinuous.

I know you've suggested me to get a finite subcover but I've tried to to that before and I didn't know how to get the appropiate ##\delta##, I am still thinking how could I solve it without using the Lebesgue number.

pasmith
Homework Helper
Let ##\epsilon>0##, I know that for each ##x \in X##, there exists ##\delta_x##: if ##y \in B_X(x,\delta_x) \implies \forall f \in \mathcal F d_Y(f(x),f(y))<\frac {\epsilon} {2}##. If ##y,z \in B(x,\delta_x) \implies d_Y(f(y),f(z)<\epsilon##.
So far so good.

I know you've suggested me to get a finite subcover but I've tried to to that before and I didn't know how to get the appropiate ##\delta##, I am still thinking how could I solve it without using the Lebesgue number.
In my earlier post, I suggested that you look at a finite subcover obtained from the open cover $\{ B(x, r(x)) : x \in X\}$. Thus there exists a finite $Z \subset X$ such that $\{ B(z, r(z)) : z \in Z \}$ is an open cover of $X$.

The entire point of the construction is that we obtain a finite set of radii of covering balls, which allows us to take
$$\delta = \min \{ r(z) : z \in Z \} > 0$$
and it remains to choose $r(z) > 0$.

If $x \in B(z,r(z))$ and $d_X(x,y) < \delta \leq r(z)$ then the triangle inequality will give a bound on $d_X(y,z)$ in terms of $r(z)$. Your aim is to ensure that $d_X(y,z) < \delta_z$.

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