Definition (Banach space):
A Banach space is a complete normed space.
Proposition (series criterion for Banach spaces):
Let be a normed space with norm . Then is a Banach space if and only if
- implies that exists in ,
whenever is a sequence in .
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
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
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.