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 ${\displaystyle Y}$.

Definitions

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be locally convex topological vector spaces. Their projective tensor product ${\displaystyle X\otimes _{\pi }Y}$ is the unique locally convex topological vector space with underlying vector space ${\displaystyle X\otimes Y}$ having the following universal property:^[1]

For any locally convex topological vector space ${\displaystyle Z}$, if ${\displaystyle \Phi _{Z}}$ is the canonical map from the vector space of bilinear maps ${\displaystyle X\times Y\to Z}$ to the vector space of linear maps ${\displaystyle X\otimes Y\to Z}$, then the image of the restriction of ${\displaystyle \Phi _{Z}}$ to the continuous bilinear maps is the space of continuous linear maps ${\displaystyle X\otimes _{\pi }Y\to Z}$.

When the topologies of ${\displaystyle X}$ and ${\displaystyle Y}$ are induced by seminorms, the topology of ${\displaystyle X\otimes _{\pi }Y}$ is induced by seminorms constructed from those on ${\displaystyle X}$ and ${\displaystyle Y}$ as follows. If ${\displaystyle p}$ is a seminorm on ${\displaystyle X}$, and ${\displaystyle q}$ is a seminorm on ${\displaystyle Y}$, define their tensor product ${\displaystyle p\otimes q}$ to be the seminorm on ${\displaystyle X\otimes Y}$ given by $${\displaystyle (p\otimes q)(b)=\inf _{r>0,\,b\in rW}r}$$for all ${\displaystyle b}$ in ${\displaystyle X\otimes Y}$, where ${\displaystyle W}$ is the balanced convex hull of the set ${\displaystyle \left\{x\otimes y:p(x)\leq 1,q(y)\leq 1\right\}}$. The projective topology on ${\displaystyle X\otimes Y}$ is generated by the collection of such tensor products of the seminorms on ${\displaystyle X}$ and ${\displaystyle Y}$.^[2][1]When ${\displaystyle X}$ and ${\displaystyle Y}$ are normed spaces, this definition applied to the norms on ${\displaystyle X}$ and ${\displaystyle Y}$ gives a norm, called the projective norm, on ${\displaystyle X\otimes Y}$ which generates the projective topology.^[3]

Properties

Throughout, all spaces are assumed to be locally convex. The symbol ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$