Locally convex topological vector space Guide, Meaning , Facts, Information and Description
In functional analysis, a topological vector space is called locally convex if its topology is defined by a set of convex neighborhoodss of 0. Every normed space is locally convex, since the triangle inequality ensures that all ballss are convex.More formally, a locally convex topological vector space (or locally convex space) is a topological vector space with the following local convexity condition: there exists a base of neighbourhoods of 0 consisting of convex sets. Equivalently, the topology is that defined by a family of semi-norms. Although such a space need not be Hausdorff, this is often also assumed.
Every Banach space is a locally convex space, and much of the theory of locally convex spaces generalises parts of the theory of Banach spaces. Indeed, local convexity is a generalisation of normable strong enough for the Hahn-Banach theorem to hold, giving a sufficiently rich theory of continuous linear functionals.
Many examples of locally convex topological vector spaces are described in the topological vector space article. On the other hand, Lp spaces for are not locally convex.
This is an Article on Locally convex topological vector space. Page Contains Information, Facts Details or Explanation Guide About Locally convex topological vector space