This is an excersice in Conway's 《A Course in Functional Analysis》.
Let $\tau:[0,1]\rightarrow [0,1]$ be continuous and define $A:C[0,1]\rightarrow C[0,1]$ by $A(f)=f\circ \tau$. Give necessary and sufficient conditions on $\tau$ for $A$ to be a compact operator.
By The Arzela-Ascoli Theorem, we know a necessary and sufficient condition is $$A(Ball(C[0,1]))$$ is equicontinuous, i.e. $\forall \epsilon \gt0,x_0\in [0,1]$, there is neighbourhood $U$ of $x_0$, s.t. $$|f(\tau(x))-f(\tau(x_0))|\le \epsilon, \forall f\in Ball(C[0,1]),\forall x\in U$$.
But how can I get a sufficient and necessary condition on $\tau$ from the above sufficient and necessary condition?
Any help would be appreciated.