Series in Banach spaces

Summary

A series $\sum x_n$ of vectors in a Banach space converges when its partial sums converge in norm. It converges absolutely when the real series $\sum \norm{x_n}$ converges. Absolute convergence implies convergence, because the space is complete. In infinite dimensions the converse fails more sharply than in $\R$: there are series that converge after every rearrangement but not absolutely. Series of this kind are used everywhere in analysis, for example in the Neumann series $\sum T^n$ for an operator $T$ of norm less than $1$.