Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.
Metrizable topologies on vector spaces have been studied since their introduction in Maurice Fréchet's 1902 PhD thesis Sur quelques points du calcul fonctionnel (wherein the notion of a metric was first introduced).
[4][5] A notable example of a result which had to wait for the development and dissemination of general locally convex spaces (amongst other notions and results, like nets, the product topology and Tychonoff's theorem) to be proven in its full generality, is the Banach–Alaoglu theorem which Stefan Banach first established in 1932 by an elementary diagonal argument for the case of separable normed spaces[6] (in which case the unit ball of the dual is metrizable).
A topological vector space (TVS) is called locally convex if it has a neighborhood basis (that is, a local base) at the origin consisting of balanced, convex sets.
[8] Every TVS has a neighborhood basis at the origin consisting of balanced sets, but only a locally convex TVS has a neighborhood basis at the origin consisting of sets that are both balanced and convex.
ranges over the positive real numbers is a subbasis at the origin for the topology induced by
Intersections of finitely many such sets are then also convex, and since the collection of all such finite intersections is a basis at the origin it follows that the topology is locally convex in the sense of the first definition given above.
is a base of continuous seminorms for a locally convex TVS
varies over the positive real numbers, is a base of neighborhoods of the origin in
ranges over all positive real numbers, forms a neighborhood basis at the origin consisting of convex open sets;[9] This forms a basis at the origin rather than merely a subbasis so that in particular, there is no need to take finite intersections of such sets.
to each element, will necessarily be a family of norms that defines this same locally convex topology.
If there exists a continuous norm on a topological vector space
form a base of convex absorbent balanced sets.
is a neighborhood base at 0 for a locally convex TVS topology on
forms a neighborhood base at the origin for a locally convex TVS topology on
into a seminormed space that carries its canonical pseudometrizable topology.
as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology.
is a topological vector space and if this convex absorbing subset
has the Hahn-Banach extension property (HBEP) if every vector subspace of
[13] The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP.
has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.
The indiscrete topology makes any vector space into a complete pseudometrizable locally convex TVS.
with the property that any linear map from it into any Hausdorff locally convex space is continuous.
The countable family of seminorms is complete and separable, so this is a Fréchet space, which is not normable.
Spaces of differentiable functions give other non-normable examples.
with the strongest locally convex topology which makes each inclusion map
that separates points and contains the constant functions (for example, the subalgebra of polynomials) is dense.
Using the seminorms, a necessary and sufficient criterion for the continuity of a linear map can be given that closely resembles the more familiar boundedness condition found for Banach spaces.
The class of all locally convex topological vector spaces forms a category with continuous linear maps as morphisms.
be a locally convex TVS whose topology is determined by a family