Seminorms and Locally Convex Spaces
نویسنده
چکیده
The first point is to describe vector spaces with topologies arising from (separating) families of semi-norms. These all turn out to be locally convex, for straightforward reasons. The second point is to check that any locally convex topological vectorspace's topology can be given by a collection of seminorms. These seminorms are made in a natural way from a local basis consisting of balanced convex neighborhoods of 0 in the vectorspace. We review the notion of completion, in which the only novelty is that we might want to allow merely pseudo-metrics rather than metrics, allowing completion with respect to semi-norms that may not be norms. Then we prove that a complete topological vectorspace with topology defined by a collection of seminorms is a projective limit of Banach spaces. We will assume that the reader is acquainted with the notions of products, coproducts, (projective) limits, and colimits (inductive limits) in a general but elementary category-theory framework. The Banach spaces obtained from vectorspaces with seminorms by completion are the building blocks from which many important topological vectorspaces are constructed, via the categorical constructs just mentioned, and by duality. (Nevertheless, completeness is too strong a condition for non-metrizable topological vectorspaces. Quasi-completeness is the appropriate general condition.) As a special case, dual spaces with weak star topologies are projective limits of finite dimensional spaces. Using an explicit construction of projective limits, we can give a complete characterization of pre-compact subsets of weak-star dual spaces, subsuming the usual version of the Banach-Alaoglu theorem.
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تاریخ انتشار 2005