A countable basis $\beta$ of a vector space $V$ over a field $F$ is a countable subset $\beta \subset V$ with the property that every element $v \in V$ can be written as an infinite series $$ v = \sum_{x \in \beta} a_x x $$ in exactly one way (where $a_x \in F$ ). We are implicitly assuming, without further comment, that the vector space $V$ has been given a topological structure or normed structure in which the above infinite sum is absolutely convergent (so that it converges to $v$ regardless of the order in which the terms are summed).

The archetypical example of a countable basis is the Fourier series of a function: every continuous real-valued periodic function $f$ on the unit circle $S^1 = \mathbb{R}/2\pi$ can be written as a Fourier series $$ f(x) = \sum_{n=0}^\infty a_n \cos(n x) + \sum_{n=1}^\infty b_n \sin(n x) $$ in exactly one way.

Note: A countable basis is a countable set, but it is not usually a basis.