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

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 ${\displaystyle Y}$ .

• If ${\displaystyle X}$  and ${\displaystyle Y}$  are both Hausdorff then so is ${\displaystyle X\otimes _{\pi }Y}$ ;^[3] if ${\displaystyle X}$  and ${\displaystyle Y}$  are Fréchet spaces then ${\displaystyle X\otimes _{\pi }Y}$  is barelled.^[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 a coarser topology than the subspace topology inherited from ${\displaystyle X\otimes _{\pi }Y}$ ).^[6]
• If ${\displaystyle E}$  and ${\displaystyle F}$  are complemented 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 the Banach spaces ${\displaystyle X}$  and ${\displaystyle Y}$ , respectively. Then ${\displaystyle E{\widehat {\otimes }}F}$  is a TVS-subspace of ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$  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 necessarily not complete^[8]). 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 ${\displaystyle B(X,Y)}$ .^[9]

Grothendieck's representation of elements in the completion edit

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 ${\displaystyle X.}$ ^[10] 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 ${\displaystyle X.}$ ^[10]

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

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

Topology of bi-bounded convergence edit

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 ${\displaystyle X{\widehat {\otimes }}_{\pi }Y}$  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 the topology of bi-bounded convergence. This topology is coarser than the strong 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 ${\displaystyle B\subseteq X{\widehat {\otimes }}Y,}$  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 are DF-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]

Strong dual and bidual edit

Let ${\displaystyle X}$  be a locally convex topological vector space and let ${\displaystyle X^{\prime }}$  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 real Lebesgue 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 ${\displaystyle L^{1}{\widehat {\otimes }}_{\pi }E}$ .^[15]

• Inductive tensor product – binary operation on topological vector spacesPages displaying wikidata descriptions as a fallback
• Injective tensor product
• Tensor product of Hilbert spaces – Tensor product space endowed with a special inner product
• Topological tensor product – Tensor product constructions for topological vector spaces

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• Nuclear space at ncatlab