absolutely summable series have Cauchy partial sums

[[concept]]

Theorem

Let $V$ be a vector space. If ${\sum_{n} v_{n}}_{n} \subset V$ is absolutely summable, then the sequence of partial sums ${\sum_{n = 1}^{m} v_{n}}_{m = 1}^{\infty}$ is Cauchy.

Proof
Exercise. Hint: first prove for $V = R$

Mentions

File	Last Modified
banach spaces have all absolutely summable series are summable	2025-05-29
Functional Analysis Lecture 2	2025-06-05

Created 2025-05-29 Last Modified 2025-05-29