Schauder basis

Summary

A Schauder basis of a Banach space is a sequence $(e_n)$ such that every vector is the sum of a unique norm-convergent series $\sum \alpha_n e_n$. It replaces the finite linear combinations of a basis by infinite series, and so it fits infinite-dimensional spaces. The unit vectors form a Schauder basis of $\ell^p$ for $1 \le p < \infty$ and of $c_0$. A Banach space with a Schauder basis is separable, but Enflo showed that not every separable Banach space has a Schauder basis.