In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS ${\displaystyle Y}$ without any need to extend definitions (such as "differentiable at a point") from real/complex-valued functions to ${\displaystyle Y}$-valued functions.

Preliminaries and notation

Throughout let ${\displaystyle X,Y,}$ and ${\displaystyle Z}$ be topological vector spaces and ${\displaystyle L:X\to Y}$ be a linear map.

• ${\displaystyle L:X\to Y}$ is a topological homomorphism or homomorphism, if it is linear, continuous, and ${\displaystyle L:X\to \operatorname {Im} L}$ is an open map, where ${\displaystyle \operatorname {Im} L=L(X)}$ has the subspace topology induced by ${\displaystyle Y}$
  • If ${\displaystyle S}$ is a subspace of ${\displaystyle X}$ then both the quotient map ${\displaystyle X\to X/S}$ and the canonical injection ${\displaystyle S\to X}$ are homomorphisms. In particular, any linear map ${\displaystyle L:X\to Y}$ can be canonically decomposed as follows: ${\displaystyle X\to X/\ker L{\overset {L_{0)){\rightarrow ))\operatorname {Im} L\to Y}$ where ${\displaystyle L_{0}(x+\ker L):=L(x)}$ defines a bijection.
• The set of continuous linear maps ${\displaystyle X\to Z}$ (resp. continuous bilinear maps ${\displaystyle X\times Y\to Z}$) will be denoted by ${\displaystyle L(X;Z)}$ (resp. ${\displaystyle B(X,Y;Z)}$) where if ${\displaystyle Z}$ is the scalar field then we may instead write ${\displaystyle L(X)}$ (resp. ${\displaystyle B(X,Y)}$).
• The set of separately continuous bilinear maps ${\displaystyle X\times Y\to Z}$ (that is, continuous in each variable when the other variable is fixed) will be denoted by ${\displaystyle {\mathcal {B))(X,Y;Z)}$ where if ${\displaystyle Z}$ is the scalar field then we may instead write ${\displaystyle {\mathcal {B))(X,Y).}$
• We will denote the continuous dual space of ${\displaystyle X}$ by ${\displaystyle X^{\prime ))$ and the algebraic dual space (which is the vector space of all linear functionals on ${\displaystyle X,}$ whether continuous or not) by ${\displaystyle X^{\#}.}$
  • To increase the clarity of the exposition, we use the common convention of writing elements of ${\displaystyle X^{\prime ))$ with a prime following the symbol (for example, ${\displaystyle x^{\prime ))$ denotes an element of ${\displaystyle X^{\prime ))$ and not, say, a derivative and the variables ${\displaystyle x}$ and ${\displaystyle x^{\prime ))$ need not be related in any way).

Notation for topologies

• ${\displaystyle \sigma \left(X,X^{\prime }\right)}$ denotes the coarsest topology on ${\displaystyle X}$ making every map in ${\displaystyle X^{\prime ))$ continuous and ${\displaystyle X_{\sigma \left(X,X^{\prime }\right)))$ or ${\displaystyle X_{\sigma ))$ denotes ${\displaystyle X}$ endowed with this topology.
• ${\displaystyle \sigma \left(X^{\prime },X\right)}$ denotes weak-* topology on ${\displaystyle X^{\prime ))$ and ${\displaystyle X_{\sigma \left(X^{\prime },X\right)))$ or ${\displaystyle X_{\sigma }^{\prime ))$ denotes ${\displaystyle X^{\prime ))$ endowed with this topology.
  • Note that every ${\displaystyle x_{0}\in X}$ induces a map ${\displaystyle X^{\prime }\to \mathbb {R} }$ defined by ${\displaystyle \lambda \mapsto \lambda \left(x_{0}\right).}$ ${\displaystyle \sigma \left(X^{\prime },X\right)}$ is the coarsest topology on X′ making all such maps continuous.
• ${\displaystyle b\left(X,X^{\prime }\right)}$ denotes the topology of bounded convergence on ${\displaystyle X}$ and ${\displaystyle X_{b\left(X,X^{\prime }\right)))$ or ${\displaystyle X_{b))$ denotes ${\displaystyle X}$ endowed with this topology.
• ${\displaystyle b\left(X^{\prime },X\right)}$ denotes the topology of bounded convergence on ${\displaystyle X^{\prime ))$ or the strong dual topology on ${\displaystyle X^{\prime ))$ and ${\displaystyle X_{b\left(X^{\prime },X\right)))$ or ${\displaystyle X_{b}^{\prime ))$ denotes ${\displaystyle X^{\prime ))$ endowed with this topology.
  • As usual, if ${\displaystyle X^{\prime ))$ is considered as a topological vector space but it has not been made clear what topology it is endowed with, then the topology will be assumed to be ${\displaystyle b\left(X^{\prime },X\right).}$
• ${\displaystyle \tau \left(X,X^{\prime }\right)}$ denotes the Mackey topology on ${\displaystyle X}$ or the topology of uniform convergence on the convex balanced weakly compact subsets of ${\displaystyle X^{\prime ))$ and ${\displaystyle X_{\tau \left(X,X^{\prime }\right)))$ or ${\displaystyle X_{\tau ))$ denotes ${\displaystyle X}$ endowed with this topology. ${\displaystyle \tau (X,X^{\prime })}$ is the finest locally convex TVS topology on ${\displaystyle X}$ whose continuous dual space is equal to ${\displaystyle X^{\prime }.}$
• ${\displaystyle \tau \left(X^{\prime },X\right)}$ denotes the Mackey topology on ${\displaystyle X^{\prime ))$ or the topology of uniform convergence on the convex balanced weakly compact subsets of ${\displaystyle X}$ and ${\displaystyle X_{\tau \left(X^{\prime },X\right)))$ or ${\displaystyle X_{\tau }^{\prime ))$ denotes ${\displaystyle X}$ endowed with this topology.
  • Note that ${\displaystyle \tau \left(X^{\prime },X\right)\subseteq b\left(X^{\prime },X\right)\subseteq \tau \left(X^{\prime },X^{\prime \prime }\right).}$^[1]
• ${\displaystyle \varepsilon \left(X,X^{\prime }\right)}$ denotes the topology of uniform convergence on equicontinuous subsets of ${\displaystyle X^{\prime ))$ and ${\displaystyle X_{\varepsilon \left(X,X^{\prime }\right)))$ or ${\displaystyle X_{\varepsilon ))$ denotes ${\displaystyle X}$ endowed with this topology.
  • If ${\displaystyle H}$ is a set of linear mappings ${\displaystyle X\to Y}$ then ${\displaystyle H}$ is equicontinuous if and only if it is equicontinuous at the origin; that is, if and only if for every neighborhood ${\displaystyle V}$ of the origin in ${\displaystyle Y,}$ there exists a neighborhood ${\displaystyle U}$ of the origin in ${\displaystyle X}$ such that ${\displaystyle \lambda (U)\subseteq V}$ for every ${\displaystyle \lambda \in H.}$
• A set ${\displaystyle H}$ of linear maps from ${\displaystyle X}$ to ${\displaystyle Y}$ is called equicontinuous if for every neighborhood ${\displaystyle V}$ of the origin in ${\displaystyle Y,}$ there exists a neighborhood ${\displaystyle U}$ of the origin in ${\displaystyle X}$ such that ${\displaystyle h(U)\subseteq V}$ for all ${\displaystyle h\in H.}$^[2]

Definition

Throughout let ${\displaystyle X}$ and ${\displaystyle Y}$ be topological vector spaces with continuous dual spaces ${\displaystyle X^{\prime ))$ and ${\displaystyle Y^{\prime }.}$ Note that almost all results described are independent of whether these vector spaces are over ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$ but to simplify the exposition we will assume that they are over the field ${\displaystyle \mathbb {C} .}$

Continuous bilinear maps as a tensor product

Despite the fact that the tensor product ${\displaystyle X\otimes Y}$ is a purely algebraic construct (its definition does not involve any topologies), the vector space ${\displaystyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$ of continuous bilinear functionals is nevertheless always a tensor product of ${\displaystyle X}$ and ${\displaystyle Y}$ (that is, ${\displaystyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)=X\otimes Y}$) when ${\displaystyle \,\otimes \,}$ is defined in the manner now described.^[3]

For every ${\displaystyle (x,y)\in X\times Y,}$ let ${\displaystyle x\otimes y}$ denote the bilinear form on ${\displaystyle X^{\prime }\times Y^{\prime ))$ defined by

$${\displaystyle (x\otimes y)\left(x^{\prime },y^{\prime }\right):=x^{\prime }(x)y^{\prime }(y).}$$

${\displaystyle x\otimes y:X_{\sigma }^{\prime }\times Y_{\sigma }^{\prime }\to \mathbb {C} }$

${\displaystyle (x,y)\in X\times Y}$

${\displaystyle x\otimes y}$

$${\displaystyle \cdot \,\otimes \,\cdot \;:\;X\times Y\to {\mathcal {B))\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$$

${\displaystyle X\otimes Y}$

${\displaystyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right).}$

${\displaystyle X_{\sigma }^{\prime }\times Y_{\sigma }^{\prime ))$

${\displaystyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)=\operatorname {span} (X\otimes Y)}$

${\displaystyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$

${\displaystyle \,\otimes \,}$

${\displaystyle X}$

${\displaystyle Y.}$

Topology

Henceforth, all topological vector spaces considered will be assumed to be locally convex. If ${\displaystyle Z}$ is any locally convex topological vector space, then ${\textstyle {\mathcal {B))\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime };Z\right)~\subseteq ~{\mathcal {B))\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right)}$^[6] and 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 {\mathcal {B))\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right)~:~b(G,H)\subseteq N\right\))$$

${\displaystyle b(G\times H)}$

${\displaystyle Z,}$

${\displaystyle {\mathcal {U))(G,H,N)}$

${\displaystyle {\mathcal {B))\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right).}$

${\displaystyle \varepsilon }$-topology

${\displaystyle \varepsilon }$

${\displaystyle \varepsilon }$

${\displaystyle {\mathcal {B))\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right)}$

${\displaystyle \varepsilon }$

${\displaystyle {\mathcal {B))_{\varepsilon }\left(X_{b}^{\prime },Y_{b}^{\prime };Z\right).}$

${\displaystyle Z}$

${\displaystyle \varepsilon }$

In the special case where ${\displaystyle Z}$ is the underlying scalar field, ${\displaystyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$ is the tensor product ${\displaystyle X\otimes Y}$ and so the topological vector space ${\displaystyle B_{\varepsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$ is called the injective tensor product of ${\displaystyle X}$ and ${\displaystyle Y}$ and it is denoted by ${\displaystyle X\otimes _{\varepsilon }Y.}$ This TVS is not necessarily complete so its completion, denoted by ${\displaystyle X{\widehat {\otimes ))_{\varepsilon }Y,}$ will be constructed. When all spaces are Hausdorff then ${\displaystyle {\mathcal {B))_{\varepsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$ is complete if and only if both ${\displaystyle X}$ and ${\displaystyle Y}$ are complete,^[8] in which case the completion ${\displaystyle X{\widehat {\otimes ))_{\varepsilon }Y}$ of ${\displaystyle B_{\varepsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$ is a vector subspace of ${\displaystyle {\mathcal {B))\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right).}$ If ${\displaystyle X}$ and ${\displaystyle Y}$ are normed spaces then so is ${\displaystyle {\mathcal {B))_{\varepsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right),}$ where ${\displaystyle {\mathcal {B))_{\varepsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$ is a Banach space if and only if this is true of both ${\displaystyle X}$ and ${\displaystyle Y.}$^[9]

Equicontinuous sets

One reason for converging on equicontinuous subsets (of all possibilities) is the following important fact:

A set of continuous linear functionals ${\displaystyle H}$ on a TVS ${\displaystyle X}$^{[note 1]} is equicontinuous if and only if it is contained in the polar of some neighborhood ${\displaystyle U}$ of the origin in ${\displaystyle X}$; that is, ${\displaystyle H\subseteq U^{\circ }.}$

A TVS's topology is completely determined by the open neighborhoods of the origin. This fact together with the bipolar theorem means that via the operation of taking the polar of a subset, the collection of all equicontinuous subsets of ${\displaystyle X^{\prime ))$ "encodes" all information about ${\displaystyle X}$'s given topology. Specifically, distinct locally convex TVS topologies on ${\displaystyle X}$ produce distinct collections of equicontinuous subsets and conversely, given any such collection of equicontinuous sets, the TVS's original topology can be recovered by taking the polar of every (equicontinuous) set in the collection. Thus through this identification, uniform convergence on the collection of equicontinuous subsets is essentially uniform convergence on the very topology of the TVS; this allows one to directly relate the injective topology with the given topologies of ${\displaystyle X}$ and ${\displaystyle Y.}$ Furthermore, the topology of a locally convex Hausdorff space ${\displaystyle X}$ is identical to the topology of uniform convergence on the equicontinuous subsets of ${\displaystyle X^{\prime }.}$^[10]

For this reason, the article now lists some properties of equicontinuous sets that are relevant for dealing with the injective tensor product. Throughout ${\displaystyle X}$ and ${\displaystyle Y}$ are any locally convex space and ${\displaystyle H}$ is a collection of linear maps from ${\displaystyle X}$ into ${\displaystyle Y.}$

• If ${\displaystyle H\subseteq L(X;Y)}$ is equicontinuous then the subspace topologies that ${\displaystyle H}$ inherits from the following topologies on ${\displaystyle L(X;Y)}$ are identical:^[11]
  1. the topology of precompact convergence;
  2. the topology of compact convergence;
  3. the topology of pointwise convergence;
  4. the topology of pointwise convergence on a given dense subset of ${\displaystyle X.}$
• An equicontinuous set ${\displaystyle H\subseteq L(X;Y)}$ is bounded in the topology of bounded convergence (that is, bounded in ${\displaystyle L_{b}(X;Y)}$).^[11] So in particular, ${\displaystyle H}$ will also bounded in every TVS topology that is coarser than the topology of bounded convergence.
• If ${\displaystyle X}$ is a barrelled space and ${\displaystyle Y}$ is locally convex then for any subset ${\displaystyle H\subseteq L(X;Y),}$ the following are equivalent:
  1. ${\displaystyle H}$ is equicontinuous;
  2. ${\displaystyle H}$ is bounded in the topology of pointwise convergence (that is, bounded in ${\displaystyle L_{\sigma }(X;Y)}$);
  3. ${\displaystyle H}$ is bounded in the topology of bounded convergence (that is, bounded in ${\displaystyle L_{b}(X;Y)}$).

In particular, to show that a set ${\displaystyle H}$ is equicontinuous it suffices to show that it is bounded in the topology of pointwise converge.^[12]

• If ${\displaystyle X}$ is a Baire space then any subset ${\displaystyle H\subseteq L(X;Y)}$ that is bounded in ${\displaystyle L_{\sigma }(X;Y)}$ is necessarily equicontinuous.^[12]
• If ${\displaystyle X}$ is separable, ${\displaystyle Y}$ is metrizable, and ${\displaystyle D}$ is a dense subset of ${\displaystyle X,}$ then the topology of pointwise convergence on ${\displaystyle D}$ makes ${\displaystyle L(X;Y)}$ metrizable so that in particular, the subspace topology that any equicontinuous subset ${\displaystyle H\subseteq L(X;Y)}$ inherits from ${\displaystyle L_{\sigma }(X;Y)}$ is metrizable.^[11]

For equicontinuous subsets of the continuous dual space ${\displaystyle X^{\prime ))$ (where ${\displaystyle Y}$ is now the underlying scalar field of ${\displaystyle X}$), the following hold:

• The weak closure of an equicontinuous set of linear functionals on ${\displaystyle X}$ is a compact subspace of ${\displaystyle X_{\sigma }^{\prime }.}$^[11]
• If ${\displaystyle X}$ is separable then every weakly closed equicontinuous subset of ${\displaystyle X_{\sigma }^{\prime ))$ is a metrizable compact space when it is given the weak topology (that is, the subspace topology inherited from ${\displaystyle X_{\sigma }^{\prime ))$).^[11]
• If ${\displaystyle X}$ is a normable space then a subset ${\displaystyle H\subseteq X^{\prime ))$ is equicontinuous if and only if it is strongly bounded (that is, bounded in ${\displaystyle X_{b}^{\prime ))$).^[11]
• If ${\displaystyle X}$ is a barrelled space then for any subset ${\displaystyle H\subseteq X^{\prime },}$ the following are equivalent:^[12]
  1. ${\displaystyle H}$ is equicontinuous;
  2. ${\displaystyle H}$ is relatively compact in the weak dual topology;
  3. ${\displaystyle H}$ is weakly bounded;
  4. ${\displaystyle H}$ is strongly bounded.

We mention some additional important basic properties relevant to the injective tensor product:

• Suppose that ${\displaystyle B:X_{1}\times X_{2}\to Y}$ is a bilinear map where ${\displaystyle X_{1))$ is a Fréchet space, ${\displaystyle X_{2))$ is metrizable, and ${\displaystyle Y}$ is locally convex. If ${\displaystyle B}$ is separately continuous then it is continuous.^[13]

Canonical identification of separately continuous bilinear maps with linear maps

The set equality ${\displaystyle L\left(X_{\sigma }^{\prime };Y_{\sigma }\right)=L\left(X_{\tau }^{\prime };Y\right)}$ always holds; that is, if ${\displaystyle u:X^{\prime }\to Y}$ is a linear map, then ${\displaystyle u:X_{\sigma \left(X^{\prime },X\right)}^{\prime }\to Y_{\sigma \left(Y,Y^{\prime }\right)))$ is continuous if and only if ${\displaystyle u:X_{\tau \left(X^{\prime },X\right)}^{\prime }\to Y}$ is continuous, where here ${\displaystyle Y}$ has its original topology.^[14]

There also exists a canonical vector space isomorphism^[14]

$${\displaystyle J:{\mathcal {B))\left(X_{\sigma \left(X^{\prime },X\right)}^{\prime },Y_{\sigma \left(Y^{\prime },Y\right)}^{\prime }\right)\to L\left(X_{\sigma \left(X^{\prime },X\right)}^{\prime };Y_{\sigma \left(Y,Y^{\prime }\right)}\right).}$$

${\displaystyle B}$

${\displaystyle X_{\sigma \left(X^{\prime },X\right)}^{\prime }\times Y_{\sigma \left(Y^{\prime },Y\right)}^{\prime ))$

${\displaystyle x^{\prime }\in X^{\prime },}$

${\displaystyle B_{x^{\prime ))\in \left(Y_{\sigma }^{\prime }\right)^{\prime ))$

$${\displaystyle B_{x^{\prime ))\left(y^{\prime }\right):=B\left(x^{\prime },y^{\prime }\right).}$$

${\displaystyle \left(Y_{\sigma }^{\prime }\right)^{\prime ))$

${\displaystyle Y}$

${\displaystyle y\mapsto }$

${\displaystyle y}$

${\displaystyle B_{x^{\prime ))}$

${\displaystyle Y,}$

${\displaystyle {\tilde {B))_{x^{\prime ))\in Y.}$

${\displaystyle {\tilde {B)):X^{\prime }\to Y}$

${\displaystyle x^{\prime }\mapsto {\tilde {B))_{x^{\prime ))}$

${\displaystyle J(B):={\tilde {B)).}$

When ${\displaystyle L\left(X_{\sigma }^{\sigma };Y_{\sigma }\right)}$ is given the topology of uniform convergence on equicontinous subsets of ${\displaystyle X^{\prime },}$ the canonical map becomes a TVS-isomorphism^[14]

$${\displaystyle J:{\mathcal {B))_{\varepsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)\to L_{\varepsilon }\left(X_{\tau }^{\prime };Y\right).}$$

${\displaystyle X\otimes _{\varepsilon }Y=B_{\varepsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$

${\displaystyle L_{\varepsilon }\left(X_{\tau }^{\prime };Y\right)}$

${\displaystyle L\left(X_{\sigma }^{\prime };Y_{\sigma }\right)}$

${\displaystyle X\otimes _{\varepsilon }Y=B_{\varepsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$

${\displaystyle J}$

${\displaystyle X_{\sigma \left(X^{\prime },X\right)}^{\prime }\to Y}$

The inclusion ${\displaystyle L\left(X_{\tau }^{\prime };Y\right)\subseteq L\left(X_{b}^{\prime };Y\right)}$ always holds. If ${\displaystyle X}$ is normed then ${\displaystyle L_{\varepsilon }\left(X_{\tau }^{\prime };Y\right)}$ is in fact a topological vector subspace of ${\displaystyle L_{b}\left(X_{b}^{\prime };Y\right).}$ And if in addition ${\displaystyle Y}$ is Banach then so is ${\displaystyle L_{b}\left(X_{b}^{\prime };Y\right)}$ (even if ${\displaystyle X}$ is not complete).^[9]

Properties

The canonical map ${\displaystyle \cdot \otimes \cdot :X\times Y\to {\mathcal {B))\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$ is always continuous^[15] and the ε-topology is always coarser than the π-topology,^[16] 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.^[15]

If ${\displaystyle X}$ and ${\displaystyle Y}$ are normed then ${\displaystyle X\otimes _{\varepsilon }Y}$ is normable in which case for all ${\displaystyle \theta \in X\otimes Y,}$ ${\displaystyle \|\theta \|_{\varepsilon }\leq \|\theta \|_{\pi }.}$^[17]

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 ${\displaystyle u\otimes v:X_{1}\otimes _{\varepsilon }X_{2}\to Y_{1}\otimes _{\varepsilon }Y_{2}.}$^[18] Moreover:

• If ${\displaystyle u}$ and ${\displaystyle v}$ are both TVS-embeddings then so is ${\displaystyle u{\widehat {\otimes ))_{\varepsilon }v:X_{1}{\widehat {\otimes ))_{\varepsilon }X_{2}\to Y_{1}{\widehat {\otimes ))_{\varepsilon }Y_{2}.}$^[19]
• 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 ${\displaystyle X_{2}{\widehat {\otimes ))_{\varepsilon }Y_{2}.}$^[20]
• 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.^[21]
• If all four spaces are normed then ${\displaystyle \|u\otimes v\|_{\varepsilon }=\|u\|\|v\|.}$^[17]

Relation to projective tensor product and nuclear spaces

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 ${\displaystyle X\times Y\to B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$ defined by sending ${\displaystyle (x,y)\in X\times Y}$ to the bilinear form ${\displaystyle x\otimes y.}$ When ${\displaystyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)=X\otimes Y}$ 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 following definition was used by Grothendieck to define nuclear spaces.^[22]

Definition 0: Let ${\displaystyle X}$ be a locally convex topological vector space. Then ${\displaystyle X}$ is nuclear if for any locally convex space ${\displaystyle Y,}$ the canonical vector space embedding ${\displaystyle X\otimes _{\pi }Y\to {\mathcal {B))_{\varepsilon }\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}$ is an embedding of TVSs whose image is dense in the codomain.

Canonical identifications of bilinear and linear maps

In this section we describe canonical identifications between spaces of bilinear and linear maps. These identifications will be used to define important subspaces and topologies (particularly those that relate to nuclear operators and nuclear spaces).

Dual spaces of the injective tensor product and its completion

Suppose that

$${\displaystyle \operatorname {In} :X\otimes _{\varepsilon }Y\to X{\widehat {\otimes ))_{\varepsilon }Y}$$

${\displaystyle X\otimes _{\varepsilon }Y}$

$${\displaystyle {}^{t}\operatorname {In} :\left(X{\widehat {\otimes ))_{\varepsilon }Y\right)_{b}^{\prime }\to \left(X\otimes _{\varepsilon }Y\right)_{b}^{\prime ))$$

${\displaystyle X\otimes _{\varepsilon }Y}$

${\displaystyle X{\widehat {\otimes ))_{\varepsilon }Y.}$

The identity map

$${\displaystyle \operatorname {Id} _{X\otimes Y}:X\otimes _{\pi }Y\to X\otimes _{\varepsilon }Y}$$

$${\displaystyle {\hat {I)):X{\widehat {\otimes ))_{\pi }Y\to X{\widehat {\otimes ))_{\varepsilon }Y.}$$

${\displaystyle X}$

${\displaystyle Y}$

${\displaystyle {\hat {I)):X{\widehat {\otimes ))_{\pi }Y\to X{\widehat {\otimes ))_{\varepsilon }Y}$

${\displaystyle X{\widehat {\otimes ))_{\varepsilon }Y}$

${\displaystyle L^{1}\left(X;Y^{\prime }\right)}$

${\displaystyle X}$

${\displaystyle Y}$

Injective tensor product of Hilbert spaces

There is a canonical map

$${\displaystyle K:X\otimes Y\to L\left(X^{\prime };Y\right)}$$

${\displaystyle z=\sum _{i=1}^{n}x_{i}\otimes y_{i))$

${\displaystyle K(z):X^{\prime }\to Y}$

$${\displaystyle K(z)\left(x^{\prime }\right):=\sum _{i=1}^{n}x^{\prime }(x_{i})y_{i}\in Y,}$$

${\displaystyle K(z):X\to Y}$

${\textstyle \sum _{i=1}^{n}x_{i}\otimes y_{i))$

${\displaystyle z.}$

$${\displaystyle K:X\otimes _{\varepsilon }Y\to L_{b}\left(X_{b}^{\prime };Y\right)}$$

${\displaystyle L_{b}\left(X_{b}^{\prime };Y\right)}$

$${\displaystyle {\hat {K)):X{\widehat {\otimes ))_{\varepsilon }Y\to L_{b}\left(X_{b}^{\prime };Y\right).}$$

When ${\displaystyle X}$ and ${\displaystyle Y}$ are Hilbert spaces then ${\displaystyle {\hat {K)):X{\widehat {\otimes ))_{\varepsilon }Y\to L_{b}\left(X_{b}^{\prime };Y\right)}$ is a TVS-embedding and isometry (when the spaces are given their usual norms) whose range is the space of all compact linear operators from ${\displaystyle X}$ into ${\displaystyle Y}$ (which is a closed vector subspace of ${\displaystyle L_{b}\left(X^{\prime };Y\right).}$ Hence ${\displaystyle X{\widehat {\otimes ))_{\varepsilon }Y}$ is identical to space of compact operators from ${\displaystyle X^{\prime ))$ into ${\displaystyle Y}$ (note the prime on ${\displaystyle X}$). The space of compact linear operators between any two Banach spaces (which includes Hilbert spaces) ${\displaystyle X}$ and ${\displaystyle Y}$ is a closed subset of ${\displaystyle L_{b}(X;Y).}$^[23]

Furthermore, the canonical map ${\displaystyle X{\widehat {\otimes ))_{\pi }Y\to X{\widehat {\otimes ))_{\varepsilon }Y}$ is injective when ${\displaystyle X}$ and ${\displaystyle Y}$ are Hilbert spaces. ^[23]

Integral forms and operators

Integral bilinear forms

Denote the identity map by

$${\displaystyle \operatorname {Id} :X\otimes _{\pi }Y\to X\otimes _{\varepsilon }Y}$$

$${\displaystyle {}^{t}\operatorname {Id} :\left(X\otimes _{\varepsilon }Y\right)_{b}^{\prime }\to \left(X\otimes _{\pi }Y\right)_{b}^{\prime ))$$

${\displaystyle \left(X\otimes _{\pi }Y\right)^{\prime ))$

${\displaystyle B(X,Y),}$

${\displaystyle X\times Y.}$

${\displaystyle X\otimes _{\varepsilon }Y}$

${\displaystyle B(X,Y),}$

${\displaystyle J(X,Y).}$

${\displaystyle J(X,Y)}$

${\displaystyle X\times Y.}$

Theorem^[24][25] — The dual ${\displaystyle J(X,Y)}$ of ${\displaystyle X{\widehat {\otimes ))_{\varepsilon }Y}$ consists of exactly those continuous bilinear forms v on ${\displaystyle X\times Y}$ that can be represented in the form of a map

$${\displaystyle b\in B(X,Y)\mapsto v(b)=\int _{S\times T}b{\big \vert }_{S\times T}\left(x^{\prime },y^{\prime }\right)\operatorname {d} \mu \left(x^{\prime },y^{\prime }\right)}$$

where ${\displaystyle S}$ and ${\displaystyle T}$ are some closed, equicontinuous subsets of ${\displaystyle X_{\sigma }^{\prime ))$ and ${\displaystyle Y_{\sigma }^{\prime },}$ respectively, and ${\displaystyle \mu }$ is a positive Radon measure on the compact set ${\displaystyle S\times T}$ with total mass ${\displaystyle \leq 1.}$ 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 ${\displaystyle S\times T.}$

Integral linear operators

Given a linear map ${\displaystyle \Lambda :X\to Y,}$ one can define a canonical bilinear form ${\displaystyle B_{\Lambda }\in Bi\left(X,Y^{\prime }\right),}$ called the associated bilinear form on ${\displaystyle X\times Y^{\prime },}$ by

$${\displaystyle B_{\Lambda }\left(x,y^{\prime }\right):=\left(y^{\prime }\circ \Lambda \right)(x).}$$

${\displaystyle \Lambda :X\to Y}$

${\displaystyle \Lambda :X\to Y}$

${\displaystyle x\in X}$

${\displaystyle y^{\prime }\in Y^{\prime }:}$

$${\displaystyle \left\langle y^{\prime },\Lambda (x)\right\rangle =\int _{A^{\prime }\times B^{\prime \prime ))\left\langle x^{\prime },x\right\rangle \left\langle y^{\prime \prime },y^{\prime }\right\rangle \operatorname {d} \mu \left(x^{\prime },y^{\prime \prime }\right)}$$

${\displaystyle A^{\prime ))$

${\displaystyle B^{\prime \prime ))$

${\displaystyle X^{\prime ))$

${\displaystyle Y^{\prime \prime },}$

${\displaystyle \mu }$

${\displaystyle \leq 1.}$

Canonical map into L(X; Y)

There is a canonical map ${\displaystyle K:X^{\prime }\otimes Y\to L(X;Y)}$ that sends ${\textstyle z=\sum _{i=1}^{n}x_{i}^{\prime }\otimes y_{i))$ to the linear map ${\displaystyle K(z):X\to Y}$ defined by ${\textstyle K(z)(x):=\sum _{i=1}^{n}x_{i}^{\prime }(x)y_{i}\in Y,}$ where it may be shown that the definition of ${\displaystyle K(z):X\to Y}$ does not depend on the particular choice of representation ${\textstyle \sum _{i=1}^{n}x_{i}^{\prime }\otimes y_{i))$ of ${\displaystyle z.}$

Examples

Space of summable families

Throughout this section we fix some arbitrary (possibly uncountable) set ${\displaystyle A,}$ a TVS ${\displaystyle X,}$ and we let ${\displaystyle {\mathcal {F))(A)}$ be the directed set of all finite subsets of ${\displaystyle A}$ directed by inclusion ${\displaystyle \subseteq .}$

Let ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A))$ be a family of elements in a TVS ${\displaystyle X}$ and for every finite subset ${\displaystyle H\subseteq A,}$ let ${\textstyle x_{H}:=\sum _{i\in H}x_{i}.}$ We call ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A))$ summable in ${\displaystyle X}$ if the limit ${\textstyle \lim _{H\in {\mathcal {F))(A)}x_{H))$ of the net ${\displaystyle \left(x_{H}\right)_{H\in {\mathcal {F))(A)))$ converges in ${\displaystyle X}$ to some element (any such element is called its sum). The set of all such summable families is a vector subspace of ${\displaystyle X^{A))$ denoted by ${\displaystyle S.}$

We now define a topology on ${\displaystyle S}$ in a very natural way. This topology turns out to be the injective topology taken from ${\displaystyle l^{1}(A){\widehat {\otimes ))_{\varepsilon }X}$ and transferred to ${\displaystyle S}$ via a canonical vector space isomorphism (the obvious one). This is a common occurrence when studying the injective and projective tensor products of function/sequence spaces and TVSs: the "natural way" in which one would define (from scratch) a topology on such a tensor product is frequently equivalent to the injective or projective tensor product topology.

Let ${\displaystyle {\mathfrak {U))}$ denote a base of convex balanced neighborhoods of 0 in ${\displaystyle X}$ and for each ${\displaystyle U\in {\mathfrak {U)),}$ let ${\displaystyle \mu _{U}:X\to \mathbb {R} }$ denote its Minkowski functional. For any such ${\displaystyle U}$ and any ${\displaystyle x=\left(x_{\alpha }\right)_{\alpha \in A}\in S,}$ let

$${\displaystyle q_{U}(x):=\sup _{x^{\prime }\in U^{\circ ))\sum _{\alpha \in A}\left|\left\langle x^{\prime },x_{\alpha }\right\rangle \right|}$$

${\displaystyle q_{U))$