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.

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

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

Let ${\displaystyle X_{b}^{\prime },Y_{b}^{\prime }}$ denote the respective dual spaces with the topology of bounded convergence. If ${\displaystyle Z}$ is a locally convex topological vector space, then ${\textstyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime };Z\right)~\subseteq ~B\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right)}$. The topology of the injective tensor product is the topology induced from a certain topology on ${\displaystyle B\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right)}$, whose basic open sets are constructed as follows. For any equicontinuous subsets ${\displaystyle G\subseteq X^{\prime }}$ and ${\displaystyle H\subseteq Y^{\prime }}$, and any neighborhood ${\displaystyle N}$ in ${\displaystyle Z}$, define $${\displaystyle {\mathcal {U}}(G,H,N)=\left\{b\in B\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right)~:~b(G\times H)\subseteq N\right\}}$$ where every set ${\displaystyle b(G\times H)}$ is bounded in ${\displaystyle Z,}$ which is necessary and sufficient for the collection of all ${\displaystyle {\mathcal {U}}(G,H,N)}$ to form a locally convex TVS topology on ${\displaystyle {\mathcal {B}}\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right).}$¹ This topology is called the ${\displaystyle \varepsilon }$-topology or injective topology. In the special case where ${\displaystyle Z=\mathbb {C} }$ is the underlying scalar field, ${\displaystyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$ is the tensor product ${\displaystyle X\otimes Y}$ as above, and the topological vector space consisting of ${\displaystyle X\otimes Y}$ with the ${\displaystyle \varepsilon }$-topology is denoted by ${\displaystyle X\otimes _{\varepsilon }Y}$, and is not necessarily complete; its completion is the injective tensor product of ${\displaystyle X}$ and ${\displaystyle Y}$ and denoted by ${\displaystyle X{\widehat {\otimes }}_{\varepsilon }Y}$.

If ${\displaystyle X}$ and ${\displaystyle Y}$ are normed spaces then ${\displaystyle X\otimes _{\varepsilon }Y}$ is normable. If ${\displaystyle X}$ and ${\displaystyle Y}$ are Banach spaces, then ${\displaystyle X{\widehat {\otimes }}_{\varepsilon }Y}$ is also. Its norm can be expressed in terms of the (continuous) duals of ${\displaystyle X}$ and ${\displaystyle Y}$. Denoting the unit balls of the dual spaces ${\displaystyle X^{*}}$ and ${\displaystyle Y^{*}}$ by ${\displaystyle B_{X^{*}}}$ and ${\displaystyle B_{Y^{*}}}$, the injective norm ${\displaystyle \|u\|_{\varepsilon }}$ of an element ${\displaystyle u\in X\otimes Y}$ is defined as $${\displaystyle \|u\|_{\varepsilon }=\sup {\big \{}{\big |}\sum _{i}\varphi (x_{i})\psi (y_{i}){\big |}:\varphi \in B_{X^{*}},\psi \in B_{Y^{*}}{\big \}}}$$ where the supremum is taken over all expressions ${\displaystyle u=\sum _{i}x_{i}\otimes y_{i}}$. Then the completion of ${\displaystyle X\otimes Y}$ under the injective norm is isomorphic as a topological vector space to ${\displaystyle X{\widehat {\otimes }}_{\varepsilon }Y}$.²

The map ${\displaystyle (x,y)\mapsto x\otimes y:X\times Y\to X\otimes _{\varepsilon }Y}$ is continuous.³

Suppose that ${\displaystyle u:X_{1}\to Y_{1}}$ and ${\displaystyle v:X_{2}\to Y_{2}}$ are two linear maps between locally convex spaces. If both ${\displaystyle u}$ and ${\displaystyle v}$ are continuous then so is their tensor product ${\displaystyle u\otimes v:X_{1}\otimes _{\varepsilon }X_{2}\to Y_{1}\otimes _{\varepsilon }Y_{2}}$. Moreover:

• If ${\displaystyle u}$ and ${\displaystyle v}$ are both TVS-embeddings then so is ${\displaystyle u{\widehat {\otimes }}_{\varepsilon }v:X_{1}{\widehat {\otimes }}_{\varepsilon }X_{2}\to Y_{1}{\widehat {\otimes }}_{\varepsilon }Y_{2}.}$
• If ${\displaystyle X_{1}}$ (resp. ${\displaystyle Y_{1}}$) is a linear subspace of ${\displaystyle X_{2}}$ (resp. ${\displaystyle Y_{2}}$) then ${\displaystyle X_{1}\otimes _{\varepsilon }Y_{1}}$ is canonically isomorphic to a linear subspace of ${\displaystyle X_{2}\otimes _{\varepsilon }Y_{2}}$ and ${\displaystyle X_{1}{\widehat {\otimes }}_{\varepsilon }Y_{1}}$ is canonically isomorphic to a linear subspace of ${\displaystyle X_{2}{\widehat {\otimes }}_{\varepsilon }Y_{2}.}$
• There are examples of ${\displaystyle u}$ and ${\displaystyle v}$ such that both ${\displaystyle u}$ and ${\displaystyle v}$ are surjective homomorphisms but ${\displaystyle u{\widehat {\otimes }}_{\varepsilon }v:X_{1}{\widehat {\otimes }}_{\varepsilon }X_{2}\to Y_{1}{\widehat {\otimes }}_{\varepsilon }Y_{2}}$ is not a homomorphism.
• If all four spaces are normed then ${\displaystyle \|u\otimes v\|_{\varepsilon }=\|u\|\|v\|.}$⁴

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

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 ${\displaystyle X\times Y\to X\otimes Y}$ separately continuous).

The space ${\displaystyle X\otimes _{\varepsilon }Y}$ is Hausdorff if and only if both ${\displaystyle X}$ and ${\displaystyle Y}$ are Hausdorff. If ${\displaystyle X}$ and ${\displaystyle Y}$ are normed then ${\displaystyle \|\theta \|_{\varepsilon }\leq \|\theta \|_{\pi }}$ for all ${\displaystyle \theta \in X\otimes Y}$, where ${\displaystyle \|\cdot \|_{\pi }}$ is the projective norm.⁵

The injective and projective topologies both figure in Grothendieck's definition of nuclear spaces.⁶

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

The dual ${\displaystyle J(X,Y)}$ of ${\displaystyle X{\widehat {\otimes }}_{\varepsilon }Y}$ consists of exactly those continuous bilinear forms ${\displaystyle v}$ on ${\displaystyle X\times Y}$ for which $${\displaystyle v(x,y)=\int _{S\times T}\varphi (x)\psi (y)\,d\mu (\varphi ,\psi )}$$ for some closed, equicontinuous subsets ${\displaystyle S}$ and ${\displaystyle T}$ of ${\displaystyle X_{\sigma }^{\prime }}$ and ${\displaystyle Y_{\sigma }^{\prime },}$ respectively, and some Radon measure ${\displaystyle \mu }$ on the compact set ${\displaystyle S\times T}$ with total mass ${\displaystyle \leq 1}$.⁷ In the case where ${\displaystyle X,Y}$ are Banach spaces, ${\displaystyle S}$ and ${\displaystyle T}$ can be taken to be the unit balls ${\displaystyle B_{X^{*}}}$ and ${\displaystyle B_{Y^{*}}}$.⁸

Furthermore, if ${\displaystyle A}$ is an equicontinuous subset of ${\displaystyle J(X,Y)}$ then the elements ${\displaystyle v\in A}$ can be represented with ${\displaystyle S\times T}$ fixed and ${\displaystyle \mu }$ running through a norm bounded subset of the space of Radon measures on ${\displaystyle S\times T.}$⁹

For ${\displaystyle X}$ a Banach space, certain constructions related to ${\displaystyle X}$ in Banach space theory can be realized as injective tensor products. Let ${\displaystyle c_{0}(X)}$ be the space of sequences of elements of ${\displaystyle X}$ converging to ${\displaystyle 0}$, equipped with the norm ${\displaystyle \|(x_{i})\|=\sup _{i}\|x_{i}\|_{X}}$. Let ${\displaystyle \ell _{1}(X)}$ be the space of unconditionally summable sequences in ${\displaystyle X}$, equipped with the norm $${\displaystyle \|(x_{i})\|=\sup {\big \{}\sum _{i=1}^{\infty }|\varphi (x_{i})|:\varphi \in B_{X^{*}}{\big \}}.}$$ Then ${\displaystyle c_{0}(X)}$ and ${\displaystyle \ell _{1}(X)}$ are Banach spaces, and isometrically ${\displaystyle c_{0}(X)\cong c_{0}{\widehat {\otimes }}_{\varepsilon }X}$ and ${\displaystyle \ell _{1}(X)\cong \ell _{1}{\widehat {\otimes }}_{\varepsilon }X}$ (where ${\displaystyle c_{0},\,\ell _{1}}$ are the classical sequence spaces).¹⁰ These facts can be generalized to the case where ${\displaystyle X}$ is a locally convex TVS.¹¹

If ${\displaystyle H}$ and ${\displaystyle K}$ are compact Hausdorff spaces, then ${\displaystyle C(H\times K)\cong C(H){\widehat {\otimes }}_{\varepsilon }C(K)}$ as Banach spaces, where ${\displaystyle C(X)}$ denotes the Banach space of continuous functions on ${\displaystyle X}$.¹¹

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

The Schwartz spaces ${\displaystyle {\mathcal {L}}\left(\mathbb {R} ^{n}\right)}$ can also be generalized to TVSs, as follows: let ${\displaystyle {\mathcal {L}}\left(\mathbb {R} ^{n};Y\right)}$ be the space of all ${\displaystyle f\in C^{\infty }\left(\mathbb {R} ^{n};Y\right)}$ such that for all pairs of polynomials ${\displaystyle P}$ and ${\displaystyle Q}$ in ${\displaystyle n}$ variables, ${\displaystyle \left\{P(x)Q\left(\partial /\partial x\right)f(x):x\in \mathbb {R} ^{n}\right\}}$ is a bounded subset of ${\displaystyle Y.}$ Topologize ${\displaystyle {\mathcal {L}}\left(\mathbb {R} ^{n};Y\right)}$ with the topology of uniform convergence over ${\displaystyle \mathbb {R} ^{n}}$ of the functions ${\displaystyle P(x)Q\left(\partial /\partial x\right)f(x),}$ as ${\displaystyle P}$ and ${\displaystyle Q}$ vary over all possible pairs of polynomials in ${\displaystyle n}$ variables. Then, ${\displaystyle {\mathcal {L}}\left(\mathbb {R} ^{n};Y\right)\cong {\mathcal {L}}\left(\mathbb {R} ^{n}\right){\widehat {\otimes }}_{\varepsilon }Y.}$¹¹