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 ${\displaystyle X}$ and ${\displaystyle Y}$, the projective topology, or π-topology, on ${\displaystyle X\otimes Y}$ is the strongest topology which makes ${\displaystyle X\otimes Y}$ a locally convex topological vector space such that the canonical map ${\displaystyle (x,y)\mapsto x\otimes y}$ (from ${\displaystyle X\times Y}$ to ${\displaystyle X\otimes Y}$) is continuous. When equipped with this topology, ${\displaystyle X\otimes Y}$ is denoted ${\displaystyle X\otimes _{\pi }Y}$ and called the projective tensor product of ${\displaystyle X}$ and ${\displaystyle Y}$.