Locally convex space - Imported Knowlpedia Content

Definition

A locally convex space is a real or complex topological vector space $E$ for which $0$ has a neighborhood basis of convex sets. Equivalently, its topology is generated by a separating family of seminorms. The neighborhoods may be chosen both convex and balanced. Under this corpus's convention, topological vector spaces are Hausdorff, so Hausdorffness is included here. Local convexity supports separation and duality while allowing topologies not defined by one norm.

Seminorm description

For a family $\mathcal P$ of seminorms on $E$ , typical basic zero-neighborhoods have the form

U(p_1,\ldots,p_n;\varepsilon) = \{x\in E:p_j(x)<\varepsilon\text{ for }1\leq j\leq n\},

where $p_1,\ldots,p_n\in\mathcal P$ and $\varepsilon>0$ . Conversely, the Minkowski gauges of suitable convex balanced zero-neighborhoods recover defining seminorms. This correspondence is formalized by the topology generated by a family of seminorms.

Separation

The seminorm topology is Hausdorff exactly when the defining family separates points: for each nonzero $x\in E$ , some $p\in\mathcal P$ satisfies $p(x)>0$ . Under this separation hypothesis, the Hahn–Banach theorem gives a rich topological dual and makes continuous linear functionals effective probes of the space.

Examples and nonexamples

Every normed vector space is locally convex, using its norm as one seminorm. Product spaces, spaces of smooth functions, and Schwartz spaces naturally require families of seminorms. By contrast, many $L^p$ -spaces with $0<p<1$ , equipped with their usual complete translation-invariant metrics, are topological vector spaces whose topologies are not locally convex.

The seminorm formulation and its use in distribution theory are developed in Trèves, Chapter 7.

References

François Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967. Elsevier publisher record. Relevant: Chapter 7.
Helmut H. Schaefer and Manfred P. Wolff, Topological Vector Spaces, 2nd ed., Springer, 1999. Springer DOI record. Relevant: Chapter II, “Locally Convex Topological Vector Spaces.”