I have been told that all vector spaces, irrespective of dimension, have a basis. That is, if $V$ is a vector space, then there exists a set $B \subset V$ such that the elements of $B$ are linearly independent, and every vector in $V$ can be written as a finite sum of elements of $V$.
It is this finite part that I am confused about. I have also, for example, been told, that "$\sin(n \pi x)$ and $\cos(m \pi x)$ forma a basis for the vector space of smooth normalizeable functions $\mathbb{R} \to \mathbb{R}$". However, not all functions can be written as a finite linear combination of such functions, thus to me, they are not a basis. Can you give an example of a basis for this space? Am I missing something?
