countable basis

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 $S^1=ℝ/2\pi$ can be written as a Fourier series

$$f(x)=\sum _{n=0}^{\mathrm{\infty }}{a}_n\cos (nx)+\sum _{n=1}^{\mathrm{\infty }}{b}_n\sin (nx)$$

in exactly one way.

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