In functional analysis and related areas of mathematics locally convex topological vector spaces or locally convex spaces are generalizations of semi normed spaces. Although such spaces are not necessarily normable they have, as with semi normed spaces, a convex local basis for 0. This condition is strong enough for the Hahn-Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

Fréchet spaces are locally convex spaces which are metrisable and complete with respect to this metric. They are generalizations of Banach spaces, which are complete vector spaces with respect to a norm.

Definition

A locally convex topological vector space (or locally convex space) is a topological vector space with the following local convexity condition: there exists a local basis for 0 consisting of convex sets.

This local convex basis can be defined by a family of seminorms in the following way:

Let P be a familiy of seminorms and be a neighbourhood for 0 in the topology induced by the seminorm p_i, then the familiy of all finite intersections of those neighbourhoods

is a local basis for 0 consisting of convex sets.

Examples

• 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.

• More generally normed space is locally convex, since the triangle inequality ensures that all balls are convex.

• The space (ω,P) of real valued sequences ω with the familiy of seminorms P defined as

• L^p spaces with are locally convex.

• Any vector space X (with or without an existing topology) can be made into a locally convex topological vector space by giving it the weakest topology making all linear functionals on X continuous. This can be seen as the weak topology defined by the algebraic dual of X.

Smooth functions

Spaces of differentiable functions give other non-normable examples. For instance, consider an open set U in Rⁿ and the set X = C^∞(U) of smooth functions f : U → R. We first define a collection of seminorms on X, and the topology will then be defined as the coarsest topology which refines the topology defined by each of the seminorms. For a compact set K and a multi-index m = (m₁, ..., m_n) we define the (K, m) semi-norm to be the supremum of the differentiation first by x₁ m₁ times, then by x₂ m₂ times and so on K. With this topology, a sequence (f_n) in X has limit f if and only if on every compact set all derivatives of f_n converge uniformly to the corresponding derivative of f. With all such semi-norms, the space X = C^∞(U) is a locally convex topological vector space, commonly denoted E(U).

Continuous functions

More abstractly, given a topological space X, the space C(X) of continuous (not necessarily bounded) functions on X can be given the topology of uniform convergence on compact sets. This topology is defined by semi-norms φ_K(f) = max { |f(x)| : x ∈ K } (as K varies over the directed set of all compact subsets of X). When X is locally compact (e.g. an open set in Rⁿ) the Stone-Weierstrass theorem applies -- any subalgebra of C(X) that separates points (e.g. polynomials) is dense.

Properties

Given a vector space X a family P of seminormes is called total if

The topology for a locally convex space is Hausdorff if and only if the family of seminorms is total.

A locally convex space is seminormable if and only if it there exists a bounded neighbourhood for zero.