Topological vector space

One of the most widely studied categories of TVSs are locally convex topological vector spaces.In this article, the scalar field of a topological vector space will be assumed to be either the complex numbersThe existence of a norm for a given topological vector space is characterized by Kolmogorov's normability criterion.Hausdorff assumption Many authors (for example, Walter Rudin), but not this page, require the topology onThe topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed below.satisfies the above two conditions but is not a filter base then it will form a neighborhood subbasis atIn general, the set of all balanced and absorbing subsets of a vector space does not satisfy the conditions of this theorem and does not form a neighborhood basis at the origin for any vector topology.Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions.These functions can then be used to prove many of the basic properties of topological vector spaces.A proof of the above theorem is given in the article on metrizable topological vector spaces.[6] so every topological vector space has a local base of absorbing and balanced sets.A TVS is pseudometrizable if and only if it has a countable neighborhood basis at the origin, or equivalent, if and only if its topology is generated by an F-seminorm.A topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of the origin.be a non-discrete locally compact topological field, for example the real or complex numbers.A topological vector space where every Cauchy sequence converges is called sequentially complete; in general, it may not be complete (in the sense that all Cauchy filters converge).The vector space operation of addition is uniformly continuous and an open map.Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus locally compact) complete pseudometrizable seminormable locally convex topological vector space.This TVS is complete, Hausdorff, and locally convex but not metrizable and consequently not normable; indeed, every neighborhood of the origin in the product topology contains lines (that is, 1-dimensional vector subspaces, which are subsets of the form(explicitly, this means that there exists a linear isomorphism between the vector spacesis a non-trivial vector space (that is, of non-zero dimension) then the discrete topology onDepending on the application additional constraints are usually enforced on the topological structure of the space.Below are some common topological vector spaces, roughly in order of increasing "niceness."This turns the dual into a locally convex topological vector space.is a TVS that is of the second category in itself (that is, a nonmeager space) then any closed convex absorbing subset ofa compact space (even if its dimension is non-zero or even infinite) and consequently also a bounded subset ofIn fact, a vector subspace of a TVS is bounded if and only if it is contained in the closure ofalso carries the trivial topology and so is itself a compact, and thus also complete, subspace (see footnote for a proof).[39] Thus, in a complete topological vector space, a closed and totally bounded subset is compact.Every finite dimensional vector subspace of a Hausdorff TVS is closed.The balanced hull of a compact (respectively, totally bounded) set has that same property.