The closure in $(L_2[0,1] \,, \,||\,||_{2})$ is compact

Question

I want to prove if this statement is either true or false but I haven't found any counter-example to prove that it's false..

The closure in $(L_2[0,1]$$,||\,||_{2})$ of equicontinuous family of continuous function on $[0,1]$ is a compact

If anyone could help it would be a lot appreciated. Thanks in advance

Answer 1

It is false. The family of constant functions furnishes a counterexample.

I think it is worth noting that a bounded set, $F$, in your space is pre-compact (the closure is compact) if and only if $\lim\limits_{x\rightarrow0}\int_0^1 |f(x+y)-f(y)|^2\,dy=0$ uniformly for $f\in F$. C.f. Dunford and Schwartz, Linear Operators, vol. 1, Theorem IV.8.20.