Projective tensor product

In functional analysis, an area of mathematics, the projective tensor product of two locally convex topological vector spaces is a natural topological vector space structure on their tensor product. Namely, given locally convex topological vector spaces and , the projective topology, or π-topology, on is the strongest topology which makes a locally convex topological vector space such that the canonical map (from to ) is continuous. When equipped with this topology, is denoted and called the projective tensor product of and . Let and be locally convex topological vector spaces. Their projective tensor product is the unique locally convex topological vector space with underlying vector space having the following universal property: For any locally convex topological vector space , if is the canonical map from the vector space of bilinear maps to the vector space of linear maps ; then the image of the restriction of to the continuous bilinear maps is the space of continuous linear maps . When the topologies of and are induced by seminorms, the topology of is induced by seminorms constructed from those on and as follows. If is a seminorm on , and is a seminorm on , define their tensor product to be the seminorm on given by for all in , where is the balanced convex hull of the set . The projective topology on is generated by the collection of such tensor products of the seminorms on and . When and are normed spaces, this definition applied to the norms on and gives a norm, called the projective norm, on which generates the projective topology. Throughout, all spaces are assumed to be locally convex. The symbol denotes the completion of the projective tensor product of and . If and are both Hausdorff then so is ; if and are Fréchet spaces then is barelled. For any two continuous linear operators and , their tensor product (as linear maps) is continuous. In general, the projective tensor product does not respect subspaces (e.g. if is a vector subspace of then the TVS has in general a coarser topology than the subspace topology inherited from ).