What would the basis of a space of continuous functions defined over a closed interval $[a,b]$ be? Also, what would the basis for a similar space with the additional constraints that $f$ is continuously differentiable and $f(a) = 0$? I reckon these 2 spaces are not isomorphic... Is anything in $\mathbb R^N$ potentially isomorphic to the continuous functions? Thanks.
Added: If the question above is not well-defined, perhaps a more explicit question might be, is the set of continuous functions on [a,b] isomorphic to $\mathbb R^n$ for some $n\in \mathbb N$? Or more generally, if given a vector space, how do I determine whether this set is isomorphic to the set of continuous functions on [a,b] ? Thanks again.