unit ball of $(C([0, 1], \mathbb{R}), \mathrm{d}\infty)$ is not compact

Question

I am trying to show that the unit ball of $(C([0, 1], \mathbb{R}), \mathrm{d}\infty)$ is not compact. thanks

Answer 1

The constant functions $f_t(x)=t$ for all $x$ give an uncountable compact subset of $C[0,1]$.

The sequence $\{1,x,x^{2},...\}$ has no convergent subsequence , so the unit ball is not compact.