Infunctional analysis, an area ofmathematics, theprojective tensor product of twolocally convex topological vector spaces is a natural topological vector space structure on theirtensor product. Namely, given locally convex topological vector spaces${\displaystyle X}$ and${\displaystyle Y}$, theprojective topology, orπ-topology, on${\displaystyle X\otimes Y}$ is thestrongest 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 atopological 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 followinguniversal property:^[1]

For any locally convex topological vector space${\displaystyle Z}$, if${\displaystyle {\mathrm{\Phi }}_Z}$ 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${\displaystyle {\mathrm{\Phi }}_Z}$ to thecontinuous bilinear maps is the space ofcontinuous linear maps${\displaystyle X{\otimes }_{\pi }Y\to Z}$.

When the topologies of${\displaystyle X}$ and${\displaystyle Y}$ are induced byseminorms, 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 theirtensor product${\displaystyle p\otimes q}$ to be the seminorm on${\displaystyle X\otimes Y}$ given by$${\displaystyle (p\otimes q)(b)=\underset{r>0,b\in rW}{inf}r}$$for all${\displaystyle b}$ in${\displaystyle X\otimes Y}$, where${\displaystyle W}$ is thebalanced convex hull of the set${\displaystyle \left\{x\otimes y:p(x)\le 1,q(y)\le 1\right\}}$. 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${\displaystyle Y}$ arenormed spaces, this definition applied to the norms on${\displaystyle X}$ and${\displaystyle Y}$ gives a norm, called theprojective norm, on${\displaystyle X\otimes Y}$ which generates the projective topology.^[3]

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

• If${\displaystyle X}$ and${\displaystyle Y}$ are bothHausdorff then so is${\displaystyle X{\otimes }_{\pi }Y}$;^[3] if${\displaystyle X}$ and${\displaystyle Y}$ areFréchet spaces then${\displaystyle X{\otimes }_{\pi }Y}$ isbarelled.^[4]
• For any two continuous linear operators${\displaystyle u_1:X_1\to Y_1}$ and${\displaystyle u_2:X_2\to Y_2}$, their tensor product (as linear maps)${\displaystyle u_1\otimes u_2:X_1{\otimes }_{\pi }{X}_2\to Y_1{\otimes }_{\pi }{Y}_2}$ is continuous.^[5]
• In general, the projective tensor product does not respect subspaces (e.g. if${\displaystyle Z}$ is a vector subspace of${\displaystyle X}$ then the TVS${\displaystyle Z{\otimes }_{\pi }Y}$ has in general acoarser topology than the subspace topology inherited from${\displaystyle X{\otimes }_{\pi }Y}$).^[6]
• If${\displaystyle E}$ and${\displaystyle F}$ arecomplemented subspaces of${\displaystyle X}$ and${\displaystyle Y,}$ respectively, then${\displaystyle E\otimes F}$ is a complemented vector subspace of${\displaystyle X{\otimes }_{\pi }Y}$ and the projective norm on${\displaystyle E{\otimes }_{\pi }F}$ is equivalent to the projective norm on${\displaystyle X{\otimes }_{\pi }Y}$ restricted to the subspace${\displaystyle E\otimes F}$. Furthermore, if${\displaystyle X}$ and${\displaystyle F}$ are complemented by projections of norm 1, then${\displaystyle E\otimes F}$ is complemented by a projection of norm 1.^[6]
• Let${\displaystyle E}$ and${\displaystyle F}$ be vector subspaces of theBanach spaces${\displaystyle X}$ and${\displaystyle Y}$, respectively. Then${\displaystyle E\hat{\otimes }F}$ is a TVS-subspace of${\displaystyle X{\hat{\otimes }}_{\pi }Y}$ if and only if every bounded bilinear form on${\displaystyle E\times F}$ extends to a continuous bilinear form on${\displaystyle X\times Y}$ with the same norm.^[7]

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 necessarilynot complete^[8]). However,${\displaystyle X{\otimes }_{\pi }Y}$ can always be linearly embedded as adense vector subspace of some complete locally convex TVS, which is generally denoted by${\displaystyle X{\hat{\otimes }}_{\pi }Y}$.

Thecontinuous dual space of${\displaystyle X{\hat{\otimes }}_{\pi }Y}$ is the same as that of${\displaystyle X{\otimes }_{\pi }Y}$, namely, the space of continuous bilinear forms${\displaystyle B(X,Y)}$.^[9]

In a Hausdorff locally convex space${\displaystyle X,}$ a sequence${\displaystyle {\left(x_i\right)}_{i=1}^{\mathrm{\infty }}}$ in${\displaystyle X}$ isabsolutely convergent if for every continuous seminorm${\displaystyle p}$ on${\displaystyle X.}$^[10] We write${\displaystyle x=\sum _{i=1}^{\mathrm{\infty }}{x}_i}$ if the sequence of partial sums${\displaystyle {\left(\sum _{i=1}^n{x}_i\right)}_{n=1}^{\mathrm{\infty }}}$ converges to${\displaystyle x}$ in${\displaystyle X.}$^[10]

The following fundamental result in the theory of topological tensor products is due toAlexander Grothendieck.^[11]

The next theorem shows that it is possible to make the representation of${\displaystyle z}$ independent of the sequences${\displaystyle {\left(x_i\right)}_{i=1}^{\mathrm{\infty }}}$ and${\displaystyle {\left(y_i\right)}_{i=1}^{\mathrm{\infty }}.}$

Let${\displaystyle {\mathfrak{B}}_X}$ and${\displaystyle {\mathfrak{B}}_Y}$ denote the families of all bounded subsets of${\displaystyle X}$ and${\displaystyle Y,}$ respectively. Since the continuous dual space of${\displaystyle X{\hat{\otimes }}_{\pi }Y}$ is the space of continuous bilinear forms${\displaystyle B(X,Y),}$ we can place on${\displaystyle B(X,Y)}$ the topology of uniform convergence on sets in${\displaystyle {\mathfrak{B}}_X\times {\mathfrak{B}}_Y,}$ which is also called thetopology of bi-bounded convergence. This topology is coarser than thestrong topology on${\displaystyle B(X,Y)}$, and in (Grothendieck 1955),Alexander Grothendieck was interested in when these two topologies were identical. This is equivalent to the problem: Given a bounded subset${\displaystyle B\subseteq X\hat{\otimes }Y,}$ do there exist bounded subsets${\displaystyle B_1\subseteq X}$ and${\displaystyle B_2\subseteq Y}$ such that${\displaystyle B}$ is a subset of the closed convex hull of${\displaystyle B_1\otimes B_2:=\{b_1\otimes b_2:b_1\in B_1,b_2\in B_2\}}$?

Grothendieck proved that these topologies are equal when${\displaystyle X}$ and${\displaystyle Y}$ are both Banach spaces or both areDF-spaces (a class of spaces introduced by Grothendieck^[13]). They are also equal when both spaces are Fréchet with one of them being nuclear.^[9]

Let${\displaystyle X}$ be a locally convex topological vector space and let${\displaystyle X^{\mathrm{\prime }}}$ be its continuous dual space. Alexander Grothendieck characterized the strong dual and bidual for certain situations:

• For${\displaystyle (X,\mathcal{A},\mu )}$ a measure space, let${\displaystyle L^1}$ be the realLebesgue space${\displaystyle L^1(\mu )}$; let${\displaystyle E}$ be a real Banach space. Let${\displaystyle L_E^1}$ be the completion of the space of simple functions${\displaystyle X\to E}$, modulo the subspace of functions${\displaystyle X\to E}$ whose pointwise norms, considered as functions${\displaystyle X\to \mathbb{R}}$, have integral${\displaystyle 0}$ with respect to${\displaystyle \mu }$. Then${\displaystyle L_E^1}$ is isometrically isomorphic to${\displaystyle L^1{\hat{\otimes }}_{\pi }E}$.^[15]

• Inductive tensor product
• Injective tensor product
• Tensor product of Hilbert spaces – Tensor product space endowed with a special inner product

• Ryan, Raymond (2002).Introduction to tensor products of Banach spaces. London New York: Springer.ISBN 1-85233-437-1.OCLC 48092184.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999).Topological Vector Spaces.GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN 978-1-4612-7155-0.OCLC 840278135.
• Trèves, François (2006) [1967].Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications.ISBN 978-0-486-45352-1.OCLC 853623322.

• Diestel, Joe (2008).The metric theory of tensor products : Grothendieck's résumé revisited. Providence, R.I: American Mathematical Society.ISBN 978-0-8218-4440-3.OCLC 185095773.
• Grothendieck, Alexander (1955). "Produits Tensoriels Topologiques et Espaces Nucléaires" [Topological Tensor Products and Nuclear Spaces].Memoirs of the American Mathematical Society Series (in French).16. Providence: American Mathematical Society.MR 0075539.OCLC 9308061.
• Grothendieck, Grothendieck (1966).Produits tensoriels topologiques et espaces nucléaires (in French). Providence: American Mathematical Society.ISBN 0-8218-1216-5.OCLC 1315788.
• Pietsch, Albrecht (1972).Nuclear locally convex spaces. Berlin, New York: Springer-Verlag.ISBN 0-387-05644-0.OCLC 539541.
• Wong (1979).Schwartz spaces, nuclear spaces, and tensor products. Berlin New York: Springer-Verlag.ISBN 3-540-09513-6.OCLC 5126158.

• Nuclear space at ncatlab

• Functional analysis
• Topological tensor products

• CS1 French-language sources (fr)