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 ${\displaystyle Y}$. It is a particular instance of a topological tensor product.

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:

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 ${\displaystyle Y}$. 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.

Throughout, all spaces are assumed to be locally convex. The symbol ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ denotes the completion of the projective tensor product of ${\displaystyle X}$ and ${\displaystyle Y}$.

In general, the space ${\displaystyle X\otimes _{\pi }Y}$ is not complete, even if both ${\displaystyle X}$ and ${\displaystyle Y}$ are complete (in fact, if ${\displaystyle X}$ and ${\displaystyle Y}$ are both infinite-dimensional Banach spaces then ${\displaystyle X\otimes _{\pi }Y}$ is necessarily not complete). However, ${\displaystyle X\otimes _{\pi }Y}$ can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$.

The continuous dual space of ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$ is the same as that of ${\displaystyle X\otimes _{\pi }Y}$, namely, the space of continuous bilinear forms ${\displaystyle B(X,Y)}$.

In a Hausdorff locally convex space ${\displaystyle X,}$ a sequence ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle X}$ is absolutely convergent if ${\displaystyle \sum _{i=1}^{\infty }p\left(x_{i}\right)<\infty }$ for every continuous seminorm ${\displaystyle p}$ on ${\displaystyle X.}$ We write ${\displaystyle x=\sum _{i=1}^{\infty }x_{i}}$ if the sequence of partial sums ${\displaystyle \left(\sum _{i=1}^{n}x_{i}\right)_{n=1}^{\infty }}$ converges to ${\displaystyle x}$ in ${\displaystyle X.}$

The following fundamental result in the theory of topological tensor products is due to Alexander Grothendieck.