injective tensor product

In functional analysis, an area of 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 spaces, as spaces of functions or sequences amount to injective tensor products of simpler spaces.

Definition

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

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

Let X^\prime_b, Y^\prime_b denote the respective dual spaces with the topology of bounded convergence. If Z is a locally convex topological vector space, then B\left(X^\prime_\sigma, Y^\prime_\sigma; Z\right)~\subseteq~ B\left(X^\prime_b, Y^\prime_b; Z\right). The topology of the injective tensor product is the topology induced from a certain topology on B\left(X^\prime_b, Y^\prime_b; Z\right), whose basic open sets are constructed as follows. For any equicontinuous subsets G \subseteq X^\prime and H \subseteq Y^\prime, and any neighborhood N in Z, define

\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\}

where every set b(G \times H) is bounded in Z, which is necessary and sufficient for the collection of all \mathcal{U}(G, H, N) to form a locally convex TVS topology on \mathcal{B}\left(X^\prime_b, Y^\prime_b; Z\right).{{sfn|Trèves|2006|pp=427–428}}{{clarify|date=January 2025}}

This topology is called the \varepsilon-topology or injective topology. In the special case where Z = \Complex is the underlying scalar field, B\left(X^\prime_\sigma, Y^\prime_\sigma\right) is the tensor product X \otimes Y as above, and the topological vector space consisting of X \otimes Y with the \varepsilon-topology is denoted by X \otimes_\varepsilon Y, and is not necessarily complete; its completion is the injective tensor product of X and Y and denoted by X \widehat{\otimes}_\varepsilon Y.

If X and Y are normed spaces then X\otimes_\varepsilon Y is normable. If X and Y are Banach spaces, then X \widehat{\otimes}_\varepsilon Y is also. Its norm can be expressed in terms of the (continuous) duals of X and Y. Denoting the unit balls of the dual spaces X^* and Y^* by B_{X^*} and B_{Y^*}, the injective norm \|u\|_\varepsilon of an element u\in X\otimes Y is defined as

\|u\|_\varepsilon = \sup\big\{\big|\sum_i \varphi(x_i)\psi(y_i)\big| : \varphi\in B_{X^*}, \psi\in B_{Y^*}\big\}

where the supremum is taken over all expressions u = \sum_i x_i\otimes y_i. Then the completion of X\otimes Y under the injective norm is isomorphic as a topological vector space to X\widehat{\otimes}_\varepsilon Y.{{sfn | Ryan | 2002 | p=45}}

Basic properties

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

Suppose that u : X_1 \to Y_1 and v : X_2 \to Y_2 are two linear maps between locally convex spaces. If both u and v are continuous then so is their tensor product u \otimes v : X_1 \otimes_\varepsilon X_2 \to Y_1 \otimes_\varepsilon Y_2. Moreover:

  • If u and v are both TVS-embeddings then so is u \widehat{\otimes}_\varepsilon v : X_1 \widehat{\otimes}_\varepsilon X_2 \to Y_1 \widehat{\otimes}_\varepsilon Y_2.
  • If X_1 (resp. Y_1) is a linear subspace of X_2 (resp. Y_2) then X_1 \otimes_\varepsilon Y_1 is canonically isomorphic to a linear subspace of X_2 \otimes_\varepsilon Y_2 and X_1 \widehat{\otimes}_\varepsilon Y_1 is canonically isomorphic to a linear subspace of X_2 \widehat{\otimes}_\varepsilon Y_2.
  • There are examples of u and v such that both u and v are surjective homomorphisms but u \widehat{\otimes}_\varepsilon v : X_1 \widehat{\otimes}_\varepsilon X_2 \to Y_1 \widehat{\otimes}_\varepsilon Y_2 is {{em|not}} a homomorphism.
  • If all four spaces are normed then \|u \otimes v\|_\varepsilon = \|u\| \|v\|.{{sfn|Trèves|2006|p=439–444}}

Relation to projective tensor product

{{Main|Projective tensor product}}

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

The injective topology is always coarser than the projective topology, which is in turn coarser than the inductive topology (the finest locally convex TVS topology making X \times Y \to X \otimes Y separately continuous).

The space X \otimes_\varepsilon Y is Hausdorff if and only if both X and Y are Hausdorff. If X and Y are normed then \|\theta\|_\varepsilon \leq \|\theta\|_{\pi} for all \theta \in X \otimes Y, where \|\cdot\|_\pi is the projective norm.{{sfn|Trèves|2006|p=434–44}}

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

Duals of injective tensor products

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

The dual J(X, Y) of X \widehat{\otimes}_\varepsilon Y consists of exactly those continuous bilinear forms v on X \times Y for which

v(x,y) = \int_{S \times T} \varphi(x)\psi(y) \,d\mu(\varphi, \psi)

for some closed, equicontinuous subsets S and T of X^{\prime}_\sigma and Y^{\prime}_\sigma, respectively, and some Radon measure \mu on the compact set S \times T with total mass \leq 1.{{sfn|Trèves|2006|pp=500–502}} In the case where X,Y are Banach spaces, S and T can be taken to be the unit balls B_{X^*} and B_{Y^*}.{{sfn|Ryan|2002|p=58}}

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

Examples

For X a Banach space, certain constructions related to X in Banach space theory can be realized as injective tensor products. Let c_0(X) be the space of sequences of elements of X converging to 0, equipped with the norm \|(x_i)\| = \sup_i \|x_i\|_X. Let \ell_1(X) be the space of unconditionally summable sequences in X, equipped with the norm

\|(x_i)\| = \sup\big\{\sum_{i=1}^\infty |\varphi(x_i)| : \varphi\in B_{X^*}\big\}.

Then c_0(X) and \ell_1(X) are Banach spaces, and isometrically c_0(X) \cong c_0 \widehat{\otimes}_\varepsilon X and \ell_1(X) \cong \ell_1 \widehat{\otimes}_\varepsilon X (where c_0, \,\ell_1 are the classical sequence spaces).{{sfn|Ryan|2002|pp=47–49}} These facts can be generalized to the case where X is a locally convex TVS.{{sfn|Trèves|2006|pp=446–451}}

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

= Spaces of differentiable functions =

{{Main|Differentiable vector-valued functions from Euclidean space}}

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

The Schwartz spaces \mathcal{L}\left(\R^n\right) can also be generalized to TVSs, as follows: let \mathcal{L}\left(\R^n; Y\right) be the space of all f \in C^{\infty}\left(\R^n; Y\right) such that for all pairs of polynomials P and Q in n variables, \left\{P(x) Q\left(\partial / \partial x\right) f(x) : x \in \R^n\right\} is a bounded subset of Y.

Topologize \mathcal{L}\left(\R^n; Y\right) with the topology of uniform convergence over \R^n of the functions P(x) Q\left(\partial / \partial x\right) f(x), as P and Q vary over all possible pairs of polynomials in n variables. Then, \mathcal{L}\left(\R^n; Y\right) \cong \mathcal{L}\left(\R^n\right) \widehat{\otimes}_\varepsilon Y.{{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}}
  • {{Trèves François Topological vector spaces, distributions and kernels}}

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}}