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	2025-11-09
Functional Analysis Lecture 2	2025-11-09

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