I am looking at a practice final and I am a bit confused by this statement I am trying to prove:
"There is a nonzero bounded linear functional on $L^\infty([0,1])$ which vanishes on the subspace $C^0([0,1])$."
[Edit: for the sake of clarity of what the comments were about, I'm leaving this paragraph:] I am used to working with $C^0(\mathbb R)$, but I am assuming that $C^0([0,1])$ is the set of bounded functions on $[0,1]$ which vanish at the endpoints. Now, the only such map I can think of (I am afraid to call it a functional, as it cannot be obtained by integrating against a $L^1$ function) is the Dirac delta at $0$.
I was hoping there would be a way to prove the existence of the functional implicitly.
[Edit: new idea.] Ok, so to apply the fact that by a Corollary of the Hanh-Banach theorem, there exists such a functional, I need to show that $C^0$ is a closed subspace. This is equivalent to having any Cauchy sequence $\{f_n\}_{n=1}^\infty\subset C^0[0,1]$ converge to a continuous function in the $L^\infty$ norm.