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.
Best Answer
Being isomorphic to $\mathbb{R}^n$ for some $n$ for a real vector space is the same as being finite dimensional. So is $C[a,b]$ (continuous functions $[a,b]\to \mathbb{R}$) finite dimensional for $a<b$? No, because it is possible to find an infinite linearly independent subset, for example, choose a sequence of functions $(f_m)_m$ where $f_m$ is nonzero in $[2^{-(m+1)},2^m]$ and zero otherwise, and then these functions form a linearly independent subset of $C[0,1]$ (obviously the case for general $a,b$ follows by scaling this example).