**analysis** (differential/integral calculus, functional analysis, topology)

metric space, normed vector space

open ball, open subset, neighbourhood

convergence, limit of a sequence

compactness, sequential compactness

continuous metric space valued function on compact metric space is uniformly continuous

…

…

In real analysis, given a sequence $a:\mathbb{N} \to \mathbb{R}$ in the real numbers, the series $\sum_{n = 0}^\infty a_n$ is **absolutely convergent** if the sequence of partial sums is a Cauchy sequence, and the sequence of partial sums of the series $\sum_{n = 0}^\infty \vert a_n \vert$ is also a Cauchy sequence.

In functional analysis, given a sequence $a:\mathbb{N} \to B$ in a Banach space $B$, the series $\sum_{n = 0}^\infty a_n$ is **absolutely convergent** if the sequence of partial sums is a Cauchy sequence in $B$, and the sequence of partial sums of the series $\sum_{n = 0}^\infty \Vert a_n \Vert$ is a Cauchy sequence in the real numbers.

- Wikipedia,
*Absolute convergence*

Created on January 4, 2023 at 19:10:17. See the history of this page for a list of all contributions to it.