Injective tensor product

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 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 $ℂ$, with continuous dual spaces $X^{\prime }$ and $Y^{\prime }.$ A subscript $\sigma$ as in $X_{\sigma }^{\prime }$ denotes the weak-* topology. Although written in terms of complex TVSs, results described generally also apply to the real case.

The vector space $B(X_{\sigma }^{\prime },Y_{\sigma }^{\prime })$ of continuous bilinear functionals $X_{\sigma }^{\prime }\times Y_{\sigma }^{\prime }\to ℂ$ 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 $f\in B(X_{\sigma }^{\prime },Y_{\sigma }^{\prime })$, given by $f(\phi ,\psi )=\phi (x)\psi (y)$. It can be shown that the map $x\otimes y\mapsto f$, extended linearly to $X\otimes Y$, is an isomorphism.

Let $X_b^{\prime },Y_b^{\prime }$ denote the respective dual spaces with the topology of bounded convergence. If $Z$ is a locally convex topological vector space, then $B(X_{\sigma }^{\prime },Y_{\sigma }^{\prime };Z)\subseteq B(X_b^{\prime },Y_b^{\prime };Z)$. The topology of the injective tensor product is the topology induced from a certain topology on $B(X_b^{\prime },Y_b^{\prime };Z)$, 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 $$𝒰(G,H,N)=\{b\in B(X_b^{\prime },Y_b^{\prime };Z):b(G\times H)\subseteq N\}$$ where every set $b(G\times H)$ is bounded in $Z,$ which is necessary and sufficient for the collection of all $𝒰(G,H,N)$ to form a locally convex TVS topology on $ℬ(X_b^{\prime },Y_b^{\prime };Z).$Template:SfnTemplate:Clarify This topology is called the $\epsilon$-topology or injective topology. In the special case where $Z=ℂ$ is the underlying scalar field, $B(X_{\sigma }^{\prime },Y_{\sigma }^{\prime })$ is the tensor product $X\otimes Y$ as above, and the topological vector space consisting of $X\otimes Y$ with the $\epsilon$-topology is denoted by $X{\otimes }_{\epsilon }Y$, and is not necessarily complete; its completion is the injective tensor product of $X$ and $Y$ and denoted by $X{\hat{\otimes }}_{\epsilon }Y$.

If $X$ and $Y$ are normed spaces then $X{\otimes }_{\epsilon }Y$ is normable. If $X$ and $Y$ are Banach spaces, then $X{\hat{\otimes }}_{\epsilon }Y$ 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 $\Vert u{\Vert }_{\epsilon }$ of an element $u\in X\otimes Y$ is defined as $$\Vert u{\Vert }_{\epsilon }=\sup \left\{\right|\sum _i\phi (x_i)\psi (y_i)|:\phi \in B_{X^{*}},\psi \in B_{Y^{*}}\}$$ 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 $X{\hat{\otimes }}_{\epsilon }Y$.Template:Sfn

Basic properties

The map $(x,y)\mapsto x\otimes y:X\times Y\to X{\otimes }_{\epsilon }Y$ is continuous.Template:Sfn

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 $u\otimes v:X_1{\otimes }_{\epsilon }{X}_2\to Y_1{\otimes }_{\epsilon }{Y}_2$. Moreover:

• If $u$ and $v$ are both TVS-embeddings then so is $u{\hat{\otimes }}_{\epsilon }v:X_1{\hat{\otimes }}_{\epsilon }{X}_2\to Y_1{\hat{\otimes }}_{\epsilon }{Y}_2.$
• If $X_1$ (resp. $Y_1$) is a linear subspace of $X_2$ (resp. $Y_2$) then $X_1{\otimes }_{\epsilon }{Y}_1$ is canonically isomorphic to a linear subspace of $X_2{\otimes }_{\epsilon }{Y}_2$ and $X_1{\hat{\otimes }}_{\epsilon }{Y}_1$ is canonically isomorphic to a linear subspace of $X_2{\hat{\otimes }}_{\epsilon }{Y}_2.$
• There are examples of $u$ and $v$ such that both $u$ and $v$ are surjective homomorphisms but $u{\hat{\otimes }}_{\epsilon }v:X_1{\hat{\otimes }}_{\epsilon }{X}_2\to Y_1{\hat{\otimes }}_{\epsilon }{Y}_2$ is Template:Em a homomorphism.
• If all four spaces are normed then $\Vert u\otimes v{\Vert }_{\epsilon }=\Vert u\Vert \Vert v\Vert .$Template:Sfn

Relation to projective tensor product

The projective topology or the $\pi$-topology is the finest locally convex topology on $B(X_{\sigma }^{\prime },Y_{\sigma }^{\prime })=X\otimes Y$ 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 $X{\otimes }_{\epsilon }Y$ is Hausdorff if and only if both $X$ and $Y$ are Hausdorff. If $X$ and $Y$ are normed then $\Vert \theta {\Vert }_{\epsilon }\le \Vert \theta {\Vert }_{\pi }$ for all $\theta \in X\otimes Y$, where $\Vert \cdot {\Vert }_{\pi }$ is the projective norm.Template:Sfn

The injective and projective topologies both figure in Grothendieck's definition of nuclear spaces.Template:Sfn

Duals of injective tensor products

The continuous dual space of $X{\otimes }_{\epsilon }Y$ 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 $X{\hat{\otimes }}_{\epsilon }Y$ consists of exactly those continuous bilinear forms $v$ on $X\times Y$ for which $$v(x,y)=\underset{S\times T}{\int }\phi (x)\psi (y)d\mu (\phi ,\psi )$$ for some closed, equicontinuous subsets $S$ and $T$ of $X_{\sigma }^{\prime }$ and $Y_{\sigma }^{\prime },$ respectively, and some Radon measure $\mu$ on the compact set $S\times T$ with total mass $\le 1$.Template:Sfn In the case where $X,Y$ are Banach spaces, $S$ and $T$ can be taken to be the unit balls $B_{X^{*}}$ and $B_{Y^{*}}$.Template:Sfn

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 $S\times T.$Template:Sfn

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 $\Vert (x_i)\Vert =\underset{i}{\sup }\Vert x_i{\Vert }_X$. Let ${\ell }_1(X)$ be the space of unconditionally summable sequences in $X$, equipped with the norm $$\Vert (x_i)\Vert =\sup \{\sum _{i=1}^{\mathrm{\infty }}|\phi (x_i)|:\phi \in B_{X^{*}}\}.$$ Then $c_0(X)$ and ${\ell }_1(X)$ are Banach spaces, and isometrically $c_0(X)\cong c_0{\hat{\otimes }}_{\epsilon }X$ and ${\ell }_1(X)\cong {\ell }_1{\hat{\otimes }}_{\epsilon }X$ (where $c_0,{\ell }_1$ are the classical sequence spaces).Template:Sfn These facts can be generalized to the case where $X$ is a locally convex TVS.Template:Sfn

If $H$ and $K$ are compact Hausdorff spaces, then $C(H\times K)\cong C(H){\hat{\otimes }}_{\epsilon }C(K)$ as Banach spaces, where $C(X)$ denotes the Banach space of continuous functions on $X$.Template:Sfn

Spaces of differentiable functions

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

The Schwartz spaces $ℒ\left({ℝ}^n\right)$ can also be generalized to TVSs, as follows: let $ℒ({ℝ}^n;Y)$ be the space of all $f\in C^{\mathrm{\infty }}({ℝ}^n;Y)$ such that for all pairs of polynomials $P$ and $Q$ in $n$ variables, $\{P(x)Q(\partial /\partial x)f(x):x\in {ℝ}^n\}$ is a bounded subset of $Y.$ Topologize $ℒ({ℝ}^n;Y)$ with the topology of uniform convergence over ${ℝ}^n$ of the functions $P(x)Q(\partial /\partial x)f(x),$ as $P$ and $Q$ vary over all possible pairs of polynomials in $n$ variables. Then, $ℒ({ℝ}^n;Y)\cong ℒ\left({ℝ}^n\right){\hat{\otimes }}_{\epsilon }Y.$Template:Sfn

References

• Template:Cite book
• Template:Schaefer Wolff Topological Vector Spaces
• Template:Trèves François Topological vector spaces, distributions and kernels

Further reading

• Template:Cite book
• Template:Grothendieck Produits Tensoriels Topologiques et Espaces Nucléaires
• Template:Cite book
• Template:Cite book
• Template:Cite book

External links

• Nuclear space at ncatlab