Compact subset of $C([0,1])$

Question

I need to prove the compactness or not compactness of:

$A=\{f\in C([0,1]):max_{x\in[0,1]}(f(x))<=1\}$

I tried using Arzela-Ascoli theorem, but had no success in it, since I don’t find enough information to conclude anything about the equicontinuity of A. Any advice?

Thank you in advance!

Answer 1

By Riesz Theorem, $\{f\in C[0,1]: \|f\|_{L^{\infty}[0,1]}\leq 1\}$ is compact if and only if $C[0,1]$ is finite dimensional.

But we know that $C[0,1]$ is not finite dimensional by looking at the linearly independent set $\{1,x,x^{2},...\}$.

Answer 2

Let $f_n(x)=x^n$. Then we can easily check that for $x\in[0,1]$, $f_n\in A$.

Further, we prove that this family of functions is not equicontinuous as in https://math.stackexchange.com/q/167522.

Thus, by Arzela-Ascoli, set is not relatively compact and thus not compact.