Let $X, Y$ be two locally convex topological vector spaces. I can tell myself a story to make $X\otimes_{\pi} Y$ and $X\otimes_{\iota} Y$ (the projective and inductive tensor products, respectively) make sense: the algebraic tensor product obeys a universal property $$\operatorname{Bilinear}(X\times Y, Z) = \operatorname{Hom}(X\otimes Y, Z).$$
In the topological world, there are two ways to interpret the left hand side: jointly continuous maps, or separately continuous maps, and these two interpretations lead to these two natural projective and inductive tensor products.
But why does one consider the injective tensor product? What is the motivation?