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Axiomatically, a locally convex space is a Hausdorff topological vector space X in which every point p has a neighborhood U which is:
convex
if x and y are points of U, the segment tx + (1 - t)y (0 ≤ t ≤ 1) lies entirely in U;
balanced
if x ∈ U, then -x ∈ U;
absorbing
for any x ∈ X, there is some real ε > 0 so that p + εx ∈ U. (Intuitively, if you puff it up enough, U covers the whole space X.)
Since the topology of a topological vector space is homogeneous from point to point (U is a neighborhood of 0 iff x + U is a neighborhood of x), it suffices to impose these three conditions on neighborhoods of 0.

Constructively, a locally convex space is built by giving a real vector space X a family of seminorms {pj} which separates the points of X, in the sense that every x in X has pj(x) ≠ 0 for some pj. Thus a Fréchet space is a topological vector space with only countably many seminorms, which turns out to be complete in the metric constructed by adding up the seminorms in a convergent series. The unit balls Bj = {x | pj(x) < 1} turn out to be the convex, balanced and absorbing neighborhoods of 0 in X.

You do not often encounter locally convex spaces which are not at least Fréchet spaces, unless you are an abstract functional analyst, operator algebraist, or some such. However, one example which occurs in partial differential equations is the space D′(M) of distributions on a noncompact manifold M. This space is topologized as the inductive limit of the spaces D′K(M) of distributions supported in K, where K ranges over all compact subsets of M. That is, it has the coarsest topology such that all the projection mappings D′(M) → D′K(M) (induced by dualizing the inclusions D(K) → D(M)) are continuous. In practice you use this construction to get around the fact that D′(M) is not a Fréchet space, by working on compact pieces of M, where it is.

For more information consult Functional analysis by Kôsaku Yosida, or Topological vector spaces by Helmut Schaefer.

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