Positive Linear Functional on $C[0,1]$ I have an exercise which seems to be missing some information. Or it could be that I really don't need that information at all. Please let me know what you think and give a solution if possible. Thank you in advance.
"A linear functional $f$ on $X = C[0,1]$ is called positive if $f(x) \geq 0$ for all nonnegative functions $x(t)$. Prove that $f \in X'$."
So here, $X'$ is the dual space where all linear bounded functionals $f:X \to \mathbb{R}$ since $X$ is just the continuous real functions. I'm thinking that the norm on $X$ is the usual maximum norm $||x(t)||_{\textrm{max}} := \textrm{max}_{t \in [0,1]}{x(t)}$ and that since $f(x(t)) \in \mathbb{R}$, the norm on it is just the absolute value norm.
So far, since we know that $f$ is a linear functional, I've been trying to show it is bounded but haven't gotten anywhere substantial.

From the given, X is a Banach space with Sup norm (from ||x|| defined as x(t) is a continuous function on compact metric space [0,1], so it is bounded). Since, f is already linear functional into another normed space R, with Uniform Boundedness Principle, f is bounded. (I believe this works...)
