# 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 $\mathbb {C}$, 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 $B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)$ of continuous bilinear functionals $X_{\sigma }^{\prime }\times Y_{\sigma }^{\prime }\to \mathbb {C}$ 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 $f\in B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)$, 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 $X_{b}^{\prime },Y_{b}^{\prime }$ denote the respective dual spaces with the topology of bounded convergence. If $Z$ is a locally convex topological vector space, then B(X$_σ$$^′$, Y$_σ$$^′$;Z) ⊆ B(X$_b$$^′$, Y$_b$$^′$;Z){\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 $B\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right)$, 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 ${\mathcal {U}}(G,H,N)=\left\{b\in B\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right)~:~b(G\times H)\subseteq N\right\}$ 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 ${\mathcal {B}}\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right).$ This topology is called the $\varepsilon$-topology or injective topology. In the special case where $Z=\mathbb {C}$ is the underlying scalar field, $B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)$ 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 $X{\widehat {\otimes }}_{\varepsilon }Y$.

## Basic properties

The map $(x,y)\mapsto x\otimes y:X\times Y\to X\otimes _{\varepsilon }Y$ is continuous.
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 not a homomorphism.
* If all four spaces are normed then $\|u\otimes v\|_{\varepsilon }=\|u\|\|v\|.$

## Relation to projective tensor product

The projective topology or the $\pi$-topology is the finest locally convex topology on $B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)=X\otimes Y$ 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.
The injective and projective topologies both figure in Grothendieck's definition of nuclear spaces.

## 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 $X_{\sigma }^{\prime }$ and $Y_{\sigma }^{\prime },$ respectively, and some Radon measure $\mu$ on the compact set $S\times T$ with total mass $\leq 1$. In the case where $X,Y$ are Banach spaces, $S$ and $T$ can be taken to be the unit balls $B_{X^{*}}$ and $B_{Y^{*}}$.
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 $S\times T.$

## 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). These facts can be generalized to the case where $X$ is a locally convex TVS.
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 $X$.

### Spaces of differentiable functions

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