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 {\displaystyle X} and Y {\displaystyle Y} be locally convex topological vector spaces over C {\displaystyle \mathbb {C} }, with continuous dual spaces X ′ {\displaystyle X^{\prime }} and Y ′ . {\displaystyle Y^{\prime }.} A subscript σ {\displaystyle \sigma } as in X σ ′ {\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 B ( X σ ′ , Y σ ′ ) {\displaystyle B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)} of continuous bilinear functionals X σ ′ × Y σ ′ → C {\displaystyle X_{\sigma }^{\prime }\times Y_{\sigma }^{\prime }\to \mathbb {C} } is isomorphic to the (vector space) tensor product X ⊗ Y {\displaystyle X\otimes Y}, as follows. For each simple tensor x ⊗ y {\displaystyle x\otimes y} in X ⊗ Y {\displaystyle X\otimes Y}, there is a bilinear map f ∈ B ( X σ ′ , Y σ ′ ) {\displaystyle f\in B\left(X_{\sigma }^{\prime },Y_{\sigma }^{\prime }\right)}, given by f ( φ , ψ ) = φ ( x ) ψ ( y ) {\displaystyle f(\varphi ,\psi )=\varphi (x)\psi (y)}. It can be shown that the map x ⊗ y ↦ f {\displaystyle x\otimes y\mapsto f}, extended linearly to X ⊗ Y {\displaystyle X\otimes Y}, is an isomorphism.

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

If X {\displaystyle X} and Y {\displaystyle Y} are normed spaces then X ⊗ ε Y {\displaystyle X\otimes _{\varepsilon }Y} is normable. If X {\displaystyle X} and Y {\displaystyle Y} are Banach spaces, then X ⊗ ^ ε Y {\displaystyle X{\widehat {\otimes }}_{\varepsilon }Y} is also. Its norm can be expressed in terms of the (continuous) duals of X {\displaystyle X} and Y {\displaystyle Y}. Denoting the unit balls of the dual spaces X ∗ {\displaystyle X^{*}} and Y ∗ {\displaystyle Y^{*}} by B X ∗ {\displaystyle B_{X^{*}}} and B Y ∗ {\displaystyle B_{Y^{*}}}, the injective norm ‖ u ‖ ε {\displaystyle \|u\|_{\varepsilon }} of an element u ∈ X ⊗ Y {\displaystyle u\in X\otimes Y} is defined as ‖ u ‖ ε = sup { | ∑ i φ ( x i ) ψ ( y i ) | : φ ∈ B X ∗ , ψ ∈ B Y ∗ } {\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 u = ∑ i x i ⊗ y i {\displaystyle u=\sum _{i}x_{i}\otimes y_{i}}. Then the completion of X ⊗ Y {\displaystyle X\otimes Y} under the injective norm is isomorphic as a topological vector space to X ⊗ ^ ε Y {\displaystyle X{\widehat {\otimes }}_{\varepsilon }Y}.

Basic properties

The map ( x , y ) ↦ x ⊗ y : X × Y → X ⊗ ε Y {\displaystyle (x,y)\mapsto x\otimes y:X\times Y\to X\otimes _{\varepsilon }Y} is continuous.

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

If u {\displaystyle u} and v {\displaystyle v} are both TVS-embeddings then so is u ⊗ ^ ε v : X 1 ⊗ ^ ε X 2 → Y 1 ⊗ ^ ε Y 2 . {\displaystyle u{\widehat {\otimes }}_{\varepsilon }v:X_{1}{\widehat {\otimes }}_{\varepsilon }X_{2}\to Y_{1}{\widehat {\otimes }}_{\varepsilon }Y_{2}.}
If X 1 {\displaystyle X_{1}} (resp. Y 1 {\displaystyle Y_{1}}) is a linear subspace of X 2 {\displaystyle X_{2}} (resp. Y 2 {\displaystyle Y_{2}}) then X 1 ⊗ ε Y 1 {\displaystyle X_{1}\otimes _{\varepsilon }Y_{1}} is canonically isomorphic to a linear subspace of X 2 ⊗ ε Y 2 {\displaystyle X_{2}\otimes _{\varepsilon }Y_{2}} and X 1 ⊗ ^ ε Y 1 {\displaystyle X_{1}{\widehat {\otimes }}_{\varepsilon }Y_{1}} is canonically isomorphic to a linear subspace of X 2 ⊗ ^ ε Y 2 . {\displaystyle X_{2}{\widehat {\otimes }}_{\varepsilon }Y_{2}.}
There are examples of u {\displaystyle u} and v {\displaystyle v} such that both u {\displaystyle u} and v {\displaystyle v} are surjective homomorphisms but u ⊗ ^ ε v : X 1 ⊗ ^ ε X 2 → Y 1 ⊗ ^ ε Y 2 {\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 ‖ u ⊗ v ‖ ε = ‖ u ‖ ‖ v ‖ . {\displaystyle \|u\otimes v\|_{\varepsilon }=\|u\|\|v\|.}

Relation to projective tensor product

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

The space X ⊗ ε Y {\displaystyle X\otimes _{\varepsilon }Y} is Hausdorff if and only if both X {\displaystyle X} and Y {\displaystyle Y} are Hausdorff. If X {\displaystyle X} and Y {\displaystyle Y} are normed then ‖ θ ‖ ε ≤ ‖ θ ‖ π {\displaystyle \|\theta \|_{\varepsilon }\leq \|\theta \|_{\pi }} for all θ ∈ X ⊗ Y {\displaystyle \theta \in X\otimes Y}, where ‖ ⋅ ‖ π {\displaystyle \|\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 ⊗ ε Y {\displaystyle X\otimes _{\varepsilon }Y} is a vector subspace of B ( X , Y ) {\displaystyle B(X,Y)}, denoted by J ( X , Y ) . {\displaystyle J(X,Y).} The elements of J ( X , Y ) {\displaystyle J(X,Y)} are called integral forms on X × Y {\displaystyle X\times Y}, a term justified by the following fact.

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

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

Examples

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

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

Spaces of differentiable functions

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

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

Notes

Ryan, Raymond (2002). Introduction to tensor products of Banach spaces. London New York: Springer. ISBN 1-85233-437-1. OCLC .
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC .
Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC .