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In [[functional analysis]], an area of [[mathematics]], the '''projective tensor product''' of two [[locally convex topological vector space]]s is a natural topological vector space structure on their [[tensor product]]. Namely, given locally convex topological vector spaces <math>X</math> and <math>Y</math>, the '''projective topology''', or '''π-topology''', on <math>X \otimes Y</math> is the [[final topology|strongest]] topology which makes <math>X \otimes Y</math> a locally convex topological vector space such that the canonical map <math>(x,y) \mapsto x \otimes y</math> (from <math>X\times Y</math> to <math>X \otimes Y</math>) is continuous. When equipped with this topology, <math>X \otimes Y</math> is denoted <math>X \otimes_\pi Y</math> and called the projective tensor product of <math>X</math> and <math>Y</math>.

== Definitions ==

Let <math>X</math> and <math>Y</math> be locally convex topological vector spaces. Their projective tensor product <math>X \otimes_\pi Y</math> is the unique locally convex topological vector space with underlying vector space <math>X \otimes Y</math> having the following [[universal property]]:{{sfn|Trèves|2006|p=438}}
:For any locally convex topological vector space <math>Z</math>, if <math>\Phi_Z</math> is the canonical map from the vector space of bilinear maps <math>X\times Y \to Z</math> to the vector space of linear maps <math>X \otimes Y \to Z</math>, then the image of the restriction of <math>\Phi_Z</math> to the ''continuous'' bilinear maps is the space of ''continuous'' linear maps <math>X \otimes_\pi Y \to Z</math>.
When the topologies of <math>X</math> and <math>Y</math> are induced by [[seminorm]]s, the topology of <math>X \otimes_\pi Y</math> is induced by seminorms constructed from those on <math>X</math> and <math>Y</math> as follows. If <math>p</math> is a seminorm on <math>X</math>, and <math>q</math> is a seminorm on <math>Y</math>, define their ''tensor product'' <math>p \otimes q</math> to be the seminorm on <math>X \otimes Y</math> given by
<math display="block">(p \otimes q)(b) = \inf_{r > 0,\, b \in r W} r</math>
for all <math>b</math> in <math>X \otimes Y</math>, where <math>W</math> is the [[Balanced set|balanced]] convex hull of the set <math>\left\{ x \otimes y : p(x) \leq 1, q(y) \leq 1 \right\}</math>. The projective topology on <math>X \otimes Y</math> is generated by the collection of such tensor products of the seminorms on <math>X</math> and <math>Y</math>.{{sfn|Trèves|2006|p=435}}{{sfn|Trèves|2006|p=438}}
When <math>X</math> and <math>Y</math> are normed spaces, this definition applied to the norms on <math>X</math> and <math>Y</math> gives a norm, called the ''projective norm'', on <math>X \otimes Y</math> which generates the projective topology.{{sfn|Trèves|2006|p=437}}

== Properties ==

Throughout, all spaces are assumed to be locally convex. The symbol <math>X \widehat{\otimes}_\pi Y</math> denotes the completion of the projective tensor product of <math>X</math> and <math>Y</math>.
* If <math>X</math> and <math>Y</math> are both [[Hausdorff space|Hausdorff]] then so is <math>X \otimes_\pi Y</math>;{{sfn|Trèves|2006|p=437}} if <math>X</math> and <math>Y</math> are [[Fréchet space]]s then <math>X \otimes_\pi Y</math> is [[Barrelled space|barelled]].{{sfn|Trèves|2006|p=445}} 
* For any two continuous linear operators <math>u_1 : X_1 \to Y_1</math> and <math>u_2 : X_2 \to Y_2</math>, their tensor product (as linear maps) <math>u_1 \otimes u_2 : X_1 \otimes_\pi X_2 \to Y_1 \otimes_\pi Y_2</math> is continuous.{{sfn|Trèves|2006|p=439}}
* In general, the projective tensor product does not respect subspaces (e.g. if <math>Z</math> is a vector subspace of <math>X</math> then the TVS <math>Z \otimes_\pi Y</math> has in general a [[comparison of topologies|coarser]] topology than the subspace topology inherited from <math>X \otimes_\pi Y</math>).{{sfn|Ryan|2002|p=18}} 
* If <math>E</math> and <math>F</math> are [[complemented subspace]]s of <math>X</math> and <math>Y,</math> respectively, then <math>E \otimes F</math> is a complemented vector subspace of <math>X \otimes_\pi Y</math> and the projective norm on <math>E \otimes_\pi F</math> is equivalent to the projective norm on <math>X \otimes_\pi Y</math> restricted to the subspace <math>E \otimes F</math>. Furthermore, if <math>X</math> and <math>F</math> are complemented by projections of norm 1, then <math>E \otimes F</math> is complemented by a projection of norm 1.{{sfn|Ryan|2002|p=18}}
* Let <math>E</math> and <math>F</math> be vector subspaces of the [[Banach space]]s <math>X</math> and <math>Y</math>, respectively. Then <math>E \widehat{\otimes} F</math> is a TVS-subspace of <math>X \widehat{\otimes}_\pi Y</math> if and only if every bounded bilinear form on <math>E \times F</math> extends to a continuous bilinear form on <math>X \times Y</math> with the same norm.{{sfn|Ryan|2002|p=24}}

== Completion ==

{{close paraphrasing|section|date=August 2023}}

In general, the space <math>X \otimes_\pi Y</math> is not complete, even if both <math>X</math> and <math>Y</math> are complete (in fact, if <math>X</math> and <math>Y</math> are both infinite-dimensional Banach spaces then <math>X \otimes_\pi Y</math> is necessarily {{em|not}} complete{{sfn|Ryan|2002|p=43}}). However, <math>X \otimes_\pi Y</math> can always be linearly embedded as a dense vector subspace of some complete locally convex TVS, which is generally denoted by <math>X \widehat{\otimes}_\pi Y</math>.

The continuous dual space of <math>X \widehat{\otimes}_\pi Y</math> is the same as that of <math>X \otimes_\pi Y</math>, namely, the space of continuous bilinear forms <math>B(X, Y)</math>.{{sfn|Schaefer|Wolff|1999|p=173}}

=== Grothendieck's representation of elements in the completion ===

In a Hausdorff locally convex space <math>X,</math> a sequence <math>\left(x_i\right)_{i=1}^{\infty}</math> in <math>X</math> is '''absolutely convergent''' if <math>\sum_{i=1}^{\infty} p \left(x_i\right) < \infty</math> for every continuous seminorm <math>p</math> on <math>X.</math>{{sfn|Schaefer|Wolff|1999|p=120}} We write <math>x = \sum_{i=1}^{\infty} x_i</math> if the sequence of partial sums <math>\left(\sum_{i=1}^n x_i\right)_{n=1}^{\infty}</math> converges to <math>x</math> in <math>X.</math>{{sfn|Schaefer|Wolff|1999|p=120}}

The following fundamental result in the theory of topological tensor products is due to [[Alexander Grothendieck]].{{sfn|Schaefer|Wolff|1999|p=94}}

{{math theorem|name=Theorem|note=|style=|math_statement=
Let <math>X</math> and <math>Y</math> be metrizable locally convex TVSs and let <math>z \in X \widehat{\otimes}_\pi Y.</math> Then <math>z</math> is the sum of an [[absolutely convergent]] series 
<math display="block">z = \sum_{i=1}^{\infty} \lambda_i x_i \otimes y_i</math>
where <math>\sum_{i=1}^{\infty}|\lambda_i|< \infty,</math> and <math>\left(x_i\right)_{i=1}^{\infty}</math> and <math>\left(y_i\right)_{i=1}^{\infty}</math> are [[null sequence]]s in <math>X</math> and <math>Y,</math> respectively.
}}

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

{{math theorem|name=Theorem{{sfn|Trèves|2006|pp=459-460}}|note=|style=|math_statement=
Let <math>X</math> and <math>Y</math> be [[Fréchet space]]s and let <math>U</math> (resp. <math>V</math>) be a balanced open neighborhood of the origin in <math>X</math> (resp. in <math>Y</math>). Let <math>K_0</math> be a compact subset of the convex balanced hull of <math>U \otimes V := \{ u \otimes v : u \in U, v \in V \}.</math> There exists a compact subset <math>K_1</math> of the unit ball in <math>\ell^1</math> and sequences <math>\left(x_i\right)_{i=1}^{\infty}</math> and <math>\left(y_i\right)_{i=1}^{\infty}</math> contained in <math>U</math> and <math>V,</math> respectively, converging to the origin such that for every <math>z \in K_0</math> there exists some <math>\left(\lambda_i\right)_{i=1}^{\infty} \in K_1</math> such that 
<math display="block">z = \sum_{i=1}^{\infty} \lambda_i x_i \otimes y_i.</math>
}}

=== Topology of bi-bounded convergence ===

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

Grothendieck proved that these topologies are equal when <math>X</math> and <math>Y</math> are both Banach spaces or both are [[DF-space]]s (a class of spaces introduced by Grothendieck{{sfn|Schaefer|Wolff|1999|p=154}}). They are also equal when both spaces are Fréchet with one of them being nuclear.{{sfn|Schaefer|Wolff|1999|p=173}}

=== Strong dual and bidual ===

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

{{math theorem|name=Theorem{{sfn|Schaefer|Wolff|1999|pp=175-176}}|note=Grothendieck|style=|math_statement=
Let <math>N</math> and <math>Y</math> be locally convex topological vector spaces with <math>N</math> [[nucalear space|nuclear]]. Assume that both <math>N</math> and <math>Y</math> are Fréchet spaces, or else that they are both [[DF-space]]s. Then, denoting strong dual spaces with a subscripted <math>b</math>:
# The strong dual of <math>N \widehat{\otimes}_\pi Y</math> can be identified with <math>N^{\prime}_b \widehat{\otimes}_\pi Y^{\prime}_b</math>; 
# The bidual of <math>N \widehat{\otimes}_\pi Y</math> can be identified with <math>N \widehat{\otimes}_\pi Y^{\prime\prime}</math>; 
# If <math>Y</math> is reflexive then <math>N \widehat{\otimes}_\pi Y</math> (and hence <math>N^{\prime}_b \widehat{\otimes}_\pi Y^{\prime}_b</math>) is a [[reflexive space]];
# Every separately continuous bilinear form on <math>N^{\prime}_b \times Y^{\prime}_b</math> is continuous; 
# Let <math>L\left(X^{\prime}_b, Y\right)</math> be the space of bounded linear maps from <math>X^{\prime}_b</math> to <math>Y</math>. Then, its strong dual can be identified with <math>N^{\prime}_b \widehat{\otimes}_\pi Y^{\prime}_b,</math> so in particular if <math>Y</math> is reflexive then so is <math>L_b\left(X^{\prime}_b, Y\right).</math>
}}

== Examples ==

*For <math>(X, \mathcal{A}, \mu)</math> a measure space, let <math>L^1</math> be the real [[Lp space|Lebesgue space]] <math>L^1(\mu)</math>; let <math>E</math> be a real Banach space. Let <math>L^1_E</math> be the completion of the space of simple functions <math>X\to E</math>, modulo the subspace of functions <math>X\to E</math> whose pointwise norms, considered as functions <math>X\to\Reals</math>, have integral <math>0</math> with respect to <math>\mu</math>. Then <math>L^1_E</math> is isometrically isomorphic to <math>L^1 \widehat{\otimes}_\pi E</math>.{{sfn|Schaefer|Wolff|1999|p=95}}

== See also ==

* {{annotated link|Inductive tensor product}}
* {{annotated link|Injective tensor product}}
* {{annotated link|Tensor product of Hilbert spaces}}
* {{annotated link|Topological tensor product}}

== Citations ==

{{reflist|group=note}}
{{reflist}}

== References ==

* {{cite book|last=Ryan|first=Raymond|title=Introduction to tensor products of Banach spaces|publisher=Springer|location=London New York|year=2002|isbn=1-85233-437-1|oclc=48092184}}
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->

== Further reading ==

* {{cite book|last=Diestel|first=Joe|title=The metric theory of tensor products : Grothendieck's résumé revisited|publisher=American Mathematical Society|location=Providence, R.I|year=2008|isbn=978-0-8218-4440-3|oclc=185095773}}
* {{Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires}} 
* {{cite book|last=Grothendieck|first=Grothendieck|title=Produits tensoriels topologiques et espaces nucléaires|publisher=American Mathematical Society|location=Providence|year=1966|isbn=0-8218-1216-5|oclc=1315788|language=fr}}
* {{cite book|last=Pietsch|first=Albrecht|title=Nuclear locally convex spaces|publisher=Springer-Verlag|location=Berlin, New York|year=1972|isbn=0-387-05644-0|oclc=539541}}
* {{cite book|author=Wong|title=Schwartz spaces, nuclear spaces, and tensor products|publisher=Springer-Verlag|location=Berlin New York|year=1979|isbn=3-540-09513-6|oclc=5126158}}

== External links ==

* [https://ncatlab.org/nlab/show/nuclear+space Nuclear space at ncatlab]

{{Topological tensor products and nuclear spaces}}
{{Functional analysis}}

[[Category:Functional analysis]]
[[Category:Topological tensor products]]