Topological Modules/Banach spaces

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Definition (Banach space):

A Banach space is a complete normed space.

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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




so that

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.