Let $S$ be a subspace of $C([0,1])$ whose elements satisfy $\|f\|_{L^\infty}\leq K\|f\|_{L^2}$. Prove that it's finite dimensional.
The obvious observation is that the $L^\infty$ and $L^2$ norms are equivalent (since the space is finite), but I can't see how to directly exploit this. The only way I know to prove a subspace is finite dimensional is to show the unit ball is compact. Using the equivalence of the norms it then seems like it'd be sufficient to prove that a sequence with $\|f_n\|_\infty$ has a subsequence converging in the (usually weaker) $L^2$ topology. I'm not sure this is true, am I on the right track?
