Topological Vector Space

A topological vector space is a vector space over or that is also a Hausdorff space and for which vector addition and scalar multiplication are jointly continuous. Every normed space, including each Banach space and Hilbert space, is a topological vector space in its norm topology.

The choice of topology determines the meaning of convergence of functions. For example, on a measure space , convergence in the norm of means

(1)

It implies convergence of the integrals, but the converse need not hold. The equivalence classes of Lebesgue integrable functions form the Banach space , the case of the L^p-spaces.

For pointwise convergence, let be a nonempty set and let or . On the vector space of all functions , each defines a seminorm

(2)

These seminorms generate the product topology, in which iff for every (Tao, p. 119). If is finite, this topology is induced by the supremum norm . If is infinite, no norm induces this topology on the full function space . A basic neighborhood of zero restricts only finitely many coordinates and contains for every , for some nonzero function supported at an unrestricted point. Such a neighborhood cannot be contained in a norm unit ball.

For a topological space , the topology of uniform convergence on compact sets on , the vector space of scalar-valued continuous functions, is generated by the seminorms

(3)

where ranges over the nonempty compact subsets of (Tao, pp. 119-120).