The Structure of Banach Spaces
This is a rather brief chapter in which we establish two general structural results about Banach spaces: One for arbitrary Banach spaces over a discretely valued field K and another one for countably generated Banach spaces over an arbitrary field K. In this book a Banach space is a complete locally convex vector space whose topology can be defined by a norm. In other words the norm is not considered to be part of the structure. Correspondingly the rich metric theory of Banach spaces is outside the scope of this book. Some part of it is present in the proofs of the two structure theorems, though. A surprising and at first disappointing consequence is the fact that over a spherically complete field K there are no infinite dimensional reflexive Banach spaces. This is probably the main reason why many applications of nonarchimedean analysis focus on different and more complicated classes of locally convex vector spaces.
KeywordsBanach Space Structure Theorem Vector Subspace Topological Isomorphism Generate Banach Space
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