Knowlpedia

Theorem

Absolutely convergent series in a Banach space

Summability of the norms ensures convergence of a vector-valued series in a complete normed space.

Statement

If XX is a and ∑n=0∞∥xn∥X<∞\sum_{n=0}^\infty\|x_n\|_X<\infty, then ∑nxn\sum_nx_n converges in XX. Moreover,

∥∑n=N∞xn∥X≤∑n=N∞∥xn∥X.\left\|\sum_{n=N}^\infty x_n\right\|_X \le\sum_{n=N}^\infty\|x_n\|_X.
Proof and operator series

The triangle inequality bounds differences of partial sums by tails of the scalar series, so the partial sums are Cauchy. Completeness supplies the limit, and continuity of the norm gives the tail bound. In particular this applies in a Banach space of bounded operators. Convergence can follow from summable bounds on powers of an operator even when its first power has norm at least one.