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In mathematics, the '''injective tensor product''' is a particular [[topological tensor product]], a [[topological vector space]] (TVS) formed by equipping the tensor product of the underlying vector spaces of two TVSs with a compatible topology. It was introduced by [[Alexander Grothendieck]] and used by him to define [[nuclear spaces]]. Injective tensor products have applications outside of nuclear spaces: as described below, many constructions of TVSs, and in particular [[Banach space]]s, as spaces of functions or sequences amount to injective tensor products of simpler spaces.

==Definition==

Let <math>X</math> and <math>Y</math> be locally convex [[topological vector space]]s over <math>\Complex</math>, with continuous dual spaces <math>X^\prime</math> and <math>Y^\prime.</math> A subscript <math>\sigma</math> as in <math>X^\prime_\sigma</math> denotes the [[weak-* topology]]. Although written in terms of complex TVSs, results described generally also apply to the real case.

The vector space <math>B\left(X^\prime_\sigma, Y^\prime_\sigma\right)</math> of continuous bilinear functionals <math>X^\prime_\sigma \times Y^\prime_\sigma \to \Complex</math> is isomorphic to the (vector space) [[tensor product]] <math>X \otimes Y</math>, as follows. For each simple tensor <math>x \otimes y</math> in <math>X \otimes Y</math>, there is a bilinear map <math>f\in B\left(X^\prime_\sigma, Y^\prime_\sigma\right)</math>, given by <math>f(\varphi,\psi) = \varphi(x)\psi(y)</math>. It can be shown that the map <math>x\otimes y\mapsto f</math>, extended linearly to <math>X\otimes Y</math>, is an isomorphism.

Let <math>X^\prime_b, Y^\prime_b</math> denote the respective dual spaces with the [[Topologies on spaces of linear maps#Topology of bounded convergence|topology of bounded convergence]]. If <math>Z</math> is a locally convex topological vector space, then <math display=inline>B\left(X^\prime_\sigma, Y^\prime_\sigma; Z\right)~\subseteq~ B\left(X^\prime_b, Y^\prime_b; Z\right)</math>. The topology of the injective tensor product is the topology [[Subspace topology|induced from]] a certain topology on <math>B\left(X^\prime_b, Y^\prime_b; Z\right)</math>, whose basic open sets are constructed as follows. For any [[Equicontinuity|equicontinuous]] subsets <math>G \subseteq X^\prime</math> and <math>H \subseteq Y^\prime</math>, and any neighborhood <math>N</math> in <math>Z</math>, define 
<math display=block>\mathcal{U}(G, H, N) = \left\{b \in B\left(X^\prime_b, Y^\prime_b; Z\right) ~:~ b(G\times H) \subseteq N\right\}</math>
where every set <math>b(G \times H)</math> is bounded in <math>Z,</math> which is necessary and sufficient for the collection of all <math>\mathcal{U}(G, H, N)</math> to form a locally convex TVS topology on <math>\mathcal{B}\left(X^\prime_b, Y^\prime_b; Z\right).</math>{{sfn|Trèves|2006|pp=427–428}}{{clarify|date=January 2025}}
This topology is called the '''<math>\varepsilon</math>-topology''' or '''injective topology'''. In the special case where <math>Z = \Complex</math> is the underlying scalar field, <math>B\left(X^\prime_\sigma, Y^\prime_\sigma\right)</math> is the tensor product <math>X \otimes Y</math> as above, and the topological vector space consisting of <math>X \otimes Y</math> with the <math>\varepsilon</math>-topology is denoted by <math>X \otimes_\varepsilon Y</math>, and is not necessarily [[Complete topological vector space|complete]]; its [[Complete topological vector space#Completions|completion]] is the '''injective tensor product''' of <math>X</math> and <math>Y</math> and denoted by <math>X \widehat{\otimes}_\varepsilon Y</math>.

If <math>X</math> and <math>Y</math> are [[normed space]]s then <math>X\otimes_\varepsilon Y</math> is normable. If <math>X</math> and <math>Y</math> are [[Banach space]]s, then <math>X \widehat{\otimes}_\varepsilon Y</math> is also. Its norm can be expressed in terms of the (continuous) duals of <math>X</math> and <math>Y</math>. Denoting the unit balls of the dual spaces <math>X^*</math> and <math>Y^*</math> by <math>B_{X^*}</math> and <math>B_{Y^*}</math>, the '''injective norm''' <math>\|u\|_\varepsilon</math> of an element <math>u\in X\otimes Y</math> is defined as
<math display=block>\|u\|_\varepsilon = \sup\big\{\big|\sum_i \varphi(x_i)\psi(y_i)\big| : \varphi\in B_{X^*}, \psi\in B_{Y^*}\big\}</math>
where the supremum is taken over all expressions <math>u = \sum_i x_i\otimes y_i</math>. Then the completion of <math>X\otimes Y</math> under the injective norm is isomorphic as a topological vector space to <math>X\widehat{\otimes}_\varepsilon Y</math>.{{sfn | Ryan | 2002 | p=45}}

==Basic properties==

The map <math>(x,y) \mapsto x\otimes y: X\times Y\to X\otimes_\varepsilon Y</math> is continuous.{{sfn|Trèves|2006|p=434}}

Suppose that <math>u : X_1 \to Y_1</math> and <math>v : X_2 \to Y_2</math> are two linear maps between locally convex spaces. If both <math>u</math> and <math>v</math> are continuous then so is their tensor product <math>u \otimes v : X_1 \otimes_\varepsilon X_2 \to Y_1 \otimes_\varepsilon Y_2</math>. Moreover:
* If <math>u</math> and <math>v</math> are both [[TVS-embedding]]s then so is <math>u \widehat{\otimes}_\varepsilon v : X_1 \widehat{\otimes}_\varepsilon X_2 \to Y_1 \widehat{\otimes}_\varepsilon Y_2.</math>
* If <math>X_1</math> (resp. <math>Y_1</math>) is a linear subspace of <math>X_2</math> (resp. <math>Y_2</math>) then <math>X_1 \otimes_\varepsilon Y_1</math> is canonically isomorphic to a linear subspace of <math>X_2 \otimes_\varepsilon Y_2</math> and <math>X_1 \widehat{\otimes}_\varepsilon Y_1</math> is canonically isomorphic to a linear subspace of <math>X_2 \widehat{\otimes}_\varepsilon Y_2.</math>
* There are examples of <math>u</math> and <math>v</math> such that both <math>u</math> and <math>v</math> are surjective homomorphisms but <math>u \widehat{\otimes}_\varepsilon v : X_1 \widehat{\otimes}_\varepsilon X_2 \to Y_1 \widehat{\otimes}_\varepsilon Y_2</math> is {{em|not}} a homomorphism.
* If all four spaces are normed then <math>\|u \otimes v\|_\varepsilon = \|u\| \|v\|.</math>{{sfn|Trèves|2006|p=439–444}}

==Relation to projective tensor product==
{{Main|Projective tensor product}}

The '''projective topology''' or the '''<math>\pi</math>-topology''' is the [[Comparison of topologies|finest]] locally convex topology on <math>B\left(X^{\prime}_\sigma, Y^{\prime}_\sigma\right) = X \otimes Y</math> that makes continuous the canonical map <math>X \times Y \to X\otimes Y</math> defined by sending <math>(x, y) \in X \times Y</math> to the bilinear form <math>x \otimes y.</math> When <math>X \otimes Y</math> is endowed with this topology then it will be denoted by <math>X \otimes_{\pi} Y</math> and called the '''[[projective tensor product]]''' of <math>X</math> and <math>Y.</math>

The injective topology is always coarser than the projective topology, which is in turn coarser than the [[inductive tensor product|inductive topology]] (the finest locally convex TVS topology making <math>X \times Y \to X \otimes Y</math> separately continuous).

The space <math>X \otimes_\varepsilon Y</math> is Hausdorff if and only if both <math>X</math> and <math>Y</math> are Hausdorff. If <math>X</math> and <math>Y</math> are normed then <math>\|\theta\|_\varepsilon \leq \|\theta\|_{\pi}</math> for all <math>\theta \in X \otimes Y</math>, where <math>\|\cdot\|_\pi</math> is the [[Projective tensor product#Definitions|projective norm]].{{sfn|Trèves|2006|p=434–44}}

The injective and projective topologies both figure in Grothendieck's definition of [[Nuclear spaces#Definition|nuclear spaces]].{{sfn|Schaefer|Wolff|1999|p=170}}

==Duals of injective tensor products==

The continuous dual space of <math>X \otimes_\varepsilon Y</math> is a vector subspace of <math>B(X, Y)</math>, denoted by <math>J(X, Y).</math> The elements of <math>J(X, Y)</math> are called ''integral forms'' on <math>X \times Y</math>, a term justified by the following fact.

The dual <math>J(X, Y)</math> of <math>X \widehat{\otimes}_\varepsilon Y</math> consists of exactly those continuous bilinear forms <math>v</math> on <math>X \times Y</math> for which
<math display=block>v(x,y) = \int_{S \times T} \varphi(x)\psi(y) \,d\mu(\varphi, \psi)</math>
for some closed, equicontinuous subsets <math>S</math> and <math>T</math> of <math>X^{\prime}_\sigma</math> and <math>Y^{\prime}_\sigma,</math> respectively, and some [[Radon measure]] <math>\mu</math> on the compact set <math>S \times T</math> with total mass <math>\leq 1</math>.{{sfn|Trèves|2006|pp=500–502}} In the case where <math>X,Y</math> are Banach spaces, <math>S</math> and <math>T</math> can be taken to be the unit balls <math>B_{X^*}</math> and <math>B_{Y^*}</math>.{{sfn|Ryan|2002|p=58}}

Furthermore, if <math>A</math> is an equicontinuous subset of <math>J(X, Y)</math> then the elements <math>v \in A</math> can be represented with <math>S \times T</math> fixed and <math>\mu</math> running through a norm bounded subset of the space of [[Radon measure]]s on <math>S \times T.</math>{{sfn|Schaefer|Wolff|1999|p=168}}

==Examples==

For <math>X</math> a Banach space, certain constructions related to <math>X</math> in Banach space theory can be realized as injective tensor products. Let <math>c_0(X)</math> be the space of sequences of elements of <math>X</math> converging to <math>0</math>, equipped with the norm <math>\|(x_i)\| = \sup_i \|x_i\|_X</math>. Let <math>\ell_1(X)</math> be the space of [[Unconditional convergence|unconditionally summable]] sequences in <math>X</math>, equipped with the norm 
<math display=block>\|(x_i)\| = \sup\big\{\sum_{i=1}^\infty |\varphi(x_i)| : \varphi\in B_{X^*}\big\}.</math>
Then <math>c_0(X)</math> and <math>\ell_1(X)</math> are Banach spaces, and isometrically <math>c_0(X) \cong c_0 \widehat{\otimes}_\varepsilon X</math> and <math>\ell_1(X) \cong \ell_1 \widehat{\otimes}_\varepsilon X</math> (where <math>c_0, \,\ell_1</math> are the classical [[sequence space]]s).{{sfn|Ryan|2002|pp=47–49}} These facts can be generalized to the case where <math>X</math> is a locally convex TVS.{{sfn|Trèves|2006|pp=446–451}}

If <math>H</math> and <math>K</math> are compact Hausdorff spaces, then <math>C(H \times K) \cong C(H) \widehat{\otimes}_\varepsilon C(K)</math> as Banach spaces, where <math>C(X)</math> denotes the [[Continuous functions on a compact Hausdorff space|Banach space of continuous functions]] on <math>X</math>.{{sfn|Trèves|2006|pp=446–451}}

=== Spaces of differentiable functions ===
{{Main|Differentiable vector-valued functions from Euclidean space}}

Let <math>\Omega</math> be an open subset of <math>\R^n</math>, let <math>Y</math> be a complete, Hausdorff, locally convex topological vector space, and let <math>C^k(\Omega; Y)</math> be the [[Differentiable vector-valued functions from Euclidean space|space of <math>k</math>-times continuously differentiable <math>Y</math>-valued functions]]. Then <math>C^k(\Omega; Y) \cong C^k(\Omega) \widehat{\otimes}_\varepsilon Y</math>.

The [[Schwartz space]]s <math>\mathcal{L}\left(\R^n\right)</math> can also be generalized to TVSs, as follows: let <math>\mathcal{L}\left(\R^n; Y\right)</math> be the space of all <math>f \in C^{\infty}\left(\R^n; Y\right)</math> such that for all pairs of polynomials <math>P</math> and <math>Q</math> in <math>n</math> variables, <math>\left\{P(x) Q\left(\partial / \partial x\right) f(x) : x \in \R^n\right\}</math> is a bounded subset of <math>Y.</math> 
Topologize <math>\mathcal{L}\left(\R^n; Y\right)</math> with the topology of uniform convergence over <math>\R^n</math> of the functions <math>P(x) Q\left(\partial / \partial x\right) f(x),</math> as <math>P</math> and <math>Q</math> vary over all possible pairs of polynomials in <math>n</math> variables. Then, <math>\mathcal{L}\left(\R^n; Y\right) \cong \mathcal{L}\left(\R^n\right) \widehat{\otimes}_\varepsilon Y.</math>{{sfn|Trèves|2006|pp=446–451}}

==Notes==

{{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]

{{TopologicalTensorProductsAndNuclearSpaces}}

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