Inductive tensor product

The strongest locally convex topological vector space (TVS) topology on formula_1, the tensor product of two locally convex TVSs, making the canonical map formula_2 (defined by sending formula_3 to formula_4) "separately" continuous is called the inductive topology or the ι-topology. When "X ⊗ Y" is endowed with this topology then it is denoted by formula_5 and called the inductive tensor product of "X" and "Y".
Preliminaries.
Throughout let "X","Y", and "Z" be topological vector spaces and formula_6 be a linear map.
Universal property.
Suppose that "Z" is a locally convex space and that "I" is the canonical map from the space of all bilinear mappings of the form formula_16, going into the space of all linear mappings of formula_67. 
Then when the domain of "I" is restricted to formula_68 (the space of separately continuous bilinear maps) then the range of this restriction is the space formula_69 of continuous linear operators formula_70.
In particular, the continuous dual space of formula_5 is canonically isomorphic to the space formula_72, the space of separately continuous bilinear forms on formula_73.
If 𝜏 is a locally convex TVS topology on "X ⊗ Y" ("X ⊗ Y" with this topology will be denoted by formula_74), then 𝜏 is equal to the inductive tensor product topology if and only if it has the following property: 