Definition (Banach space):
A Banach space is a complete normed space.
TODO: Links
Proof: Suppose first that
is a Banach space. Then suppose that
converges, where
is a sequence in
. Then set
; we claim that
is a Cauchy sequence. Indeed, for
sufficiently large, we have
.
Hence,
also converges, because
is a Banach space.
Now suppose that for all sequences
the implication
![{\displaystyle \sum _{n=1}^{\infty }\|x_{n}\|<\infty \Rightarrow \lim _{N\to \infty }\sum _{n=1}^{N}x_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93e6fb33e1cf0f82bfd3cf1ea1a20608f181bbf0)
holds. Let then
be a Cauchy sequence in
. By the Cauchy property, choose, for all
, a number
such that
whenever
. We may assume that
, ie.
is an ascending sequence of natural numbers. Then define
and for
set
. Then
.
Moreover,
,
so that
![{\displaystyle \sum _{j=1}^{\infty }\|x_{j}\|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40b3013d7a7a91a30b9a81701b669bf65750823e)
converges as a monotonely increasing, bounded sequence. By the assumption, the sequence
converges, where
.
Thus,
is a Cauchy sequence that has a convergent subsequence and is hence convergent.