Topological Vector Spaces/Elementary constructions

Topological vector space with the initial topology Let be a vector space, let be a family of topological vector spaces and let be a family of linear functions. Then together with the initial topology induced by the is a topological vector space. We are to show that scalar multiplication and addition are continuous. But this follows immediately, since a function from a topological space to is continuous under the initial topology if and only if for all the function is continuous (see General topology/ Dual systems, the weak topology duality of vector spaces Let be vector spaces over the field . are set in duality iff there exists a bilinear function .