Algebraic Tensor Products Revisited: Axiomatic Approach
aa r X i v : . [ m a t h . F A ] J a n Bulletin of the Malaysian Mathematical Sciences Society , (2021) to appear.
ALGEBRAIC TENSOR PRODUCTS REVISITED: AXIOMATICAPPROACH
C.S. KUBRUSLY
Abstract.
This is an expository paper on tensor products where the standardapproaches for constructing concrete instances of algebraic tensor products oflinear spaces, via quotient spaces or via linear maps of bilinear maps, arereviewed by reducing them to different but isomorphic interpretations of anabstract notion, viz., the universal property, which is based on a pair of axioms. Introduction
The purpose of this paper is to offer a brief and unified review with an expositoryflavor on the common realizations of algebraic tensor products (either via quotientspace or via linear maps of bilinear maps) by reversing the presentation order . Inother words, this exposition focuses on the approach where the so-called universalproperty is taken as an axiomatic starting point, rather than as a theorem for aspecific realization . This leads to the abstract notion of algebraic tensor productsof linear spaces (the pre-crossnorm stage), where the concrete standard forms areshown to be interpretations of the axiomatic formulation.The origin of a systematic presentation of tensor products in book form datesback to Schatten’s 1950 monograph [21], where the notion of direct product oflinear spaces was given in terms of formal products (which match what its nowcalled tensor product space and single tensors ) and their symbols in the form offinite sum of formal products . This followed a Kronecker-product-like notion onfinite-dimensional spaces given in Weyl’s 1931 book [23, Chapter V] . Grothendieck’sfundamental work in the 1950’s has been unified and updated by Diestel, Fourieand Swart in 2008 [3] . See also Pisier’s 2012 exposition [17] . In Grothendieck’spioneering work, the notion of tensor product space was essentially given in termsof the dual of the linear space of bilinear forms (or, more generally, the linear spaceof linear maps of bilinear maps) . The same way of defining tensor product alsoappears in Halmos’s 1958 book on finite-dimensional vector spaces [5, Section 24](although the formal products variant is also mentioned as a possible alternative) . Another representative along this line (dual of the linear space of bilinear forms) isRyan’s 2002 book on tensor products of Banach spaces [20].On the other hand, a different but still usual approach for defining tensor productrelies on quotient spaces of free linear spaces (equivalent to the linear space of formallinear combinations) of Cartesian products of linear spaces . This has been sometimesreferred to as algebraic tensor product (although the previous approach is equallyalgebraic) . See e.g., [1, pp.22–25], [22, Section 3.4] and [19, Chapter 14] for a linearspace version, and [15, Section IX.8] and [14, Section XVI.1]) for a module version.The present paper is organized as follows . Notation, terminology and auxiliaryresults are brought together in Section 2 . This is split into three parts: formal linearcombination, quotient space, and bilinear maps . Tensor products are axiomaticallydefined in Section 3, and common properties shared by the concrete formulations
Date : July 10, 2020; revised, November 24, 2020.2020
Mathematics Subject Classification.
Keywords.
Bilinear maps, quotient spaces, tensor product. are obtained from such an abstract formulation . The usual realizations, viz., thequotient space and linear maps of bilinear maps, are individually inferred fromDefinition 3.1 and are considered in Sections 4 and 5.2.
Notation, Terminology, and Auxiliary Results
Let X and Y be linear spaces over the same field F , and let L [ X , Y ] denote thelinear space over F of all linear transformations from X to Y . For X = Y write L [ X ] = L [ X , X ], which is the algebra of all linear transformations of X into itself . The kernel N ( L ) = L − ( { } ) and range R ( L ) = L ( X ) of L ∈ L [ X , Y ] are linearmanifolds of X and Y respectively . A linear transformation L ∈ L [ X , Y ] is injectiveif N ( L ) = { } and surjective if R ( L ) = Y . By an isomorphism (or an algebraicisomorphism , or a linear-space isomorphism ) we mean an invertible (i.e., injectiveand surjective) linear transformation . Two linear spaces X and Y are isomorphic (notation X ∼ = Y ) if there is an isomorphism between them . For the particular caseof Y = F the elements of L [ X , F ] are referred to as linear functionals or linearforms , and the linear space L [ X , F ] of all linear functionals on X is referred toas the algebraic dual of X , denoted by X ♯ (i.e., X ♯ = L [ X , F ]) . For an arbitrarylinear transformation L ∈ L [ X , Y ] consider the linear transformation L ♯ ∈ L [ Y ♯ , X ♯ ]defined by L ♯ g = gL ∈ X ♯ for every g ∈ Y ♯ (i.e., ( L ♯ g )( x ) = g ( Lx ) ∈ F for every g ∈ Y ♯ and every x ∈ X — we use both notations gL or g ◦ L for composition) . This L ♯ ∈ L [ Y ♯ , X ♯ ] is the algebraic adjoint of L ∈ L [ X , Y ].The next subsections summarize not only notation and terminology, but alsoauxiliary results that will be required in Sections 4 and 5.2.1. Formal Linear Combination.
Let S be an arbitrary nonempty set and let F be a field . Consider the linear space F S of all scalar-valued functions f : S → F on S. Let S / = (cid:8) f ∈ F S : f ( S \ A ) = 0 for some A ⊆ S with A < ∞ (cid:9) of all functions f : S → F which vanish everywhere on the complement of somefinite subset A of S (which depends on f ) . This S / is a linear manifold of F S , andso is itself a linear space over F . For each s ∈ S take the characteristic function e s = χ { s } : S → F of the singleton { s } ⊆ S. As is readily verified the set S = { e s } s ∈ S is a Hamel basis for the linear space S /. Thus an arbitrary vector f ∈ S / , being a scalar-valued function taking nonzero valuesonly over a finite subset { s i } ni =1 of S , has a unique expansion with α i ∈ F : f = X ni =1 α i e s i ∈ S /. The linear space S / is called the free linear space generated by S. Since { e s } s ∈ S = S (i.e., each element s from the set S is in a one-to-one correspondence with eachfunction e s from the Hamel basis S for the function space S / ), then dim S / = S = S. This sets a natural identification ≈ such that s ≈ e s and so S ≈ S , which in turnleads to a natural identification for an arbitrary linear combination in S / , X ni =1 α i e s i ≈ X ni =1 α i s i , where P ni =1 α i s i is referred to as a formal linear combination of points s i ∈ S (al-though addition or scalar multiplication is not directly defined on the set S ), thecollection of which is the linear space of formal linear combinations from S. So any
ENSOR PRODUCTS REVISITED: AXIOMATIC APPROACH 3 function f in the linear space S / is identified with a formal linear combination ofpoints in S , and the set S that generates the free linear space S / is identified withthe Hamel basis S for S /. In this sense the set S may be regarded as a subset of S / ,and a function f in S / may be regarded as a formal linear combination . Thus write f = X ni =1 α i s i for X ni =1 α i s i ≈ f ∈ S / and S ⊂ S / for S ≈ S ⊂ S /. Quotient Space.
Let M be a linear manifold of a linear space X over a field F , let [ x ] = x + M denote the coset of x ∈ X modulo M , and let X / M stand forthe quotient space of X modulo M , which is the linear space over F of all cosets[ x ] modulo M for every x ∈ X . Consider the quotient map (or the natural quotientmap ) π : X → X / M of the linear space X onto the linear space X / M , defined by π ( x ) = [ x ] = x + M for every x ∈ X , which is a surjective linear transformation according to the usual definition of ad-dition and scalar multiplication in X / M . Proposition 2.1.
Universal Property.
Let X and Z be linear spaces over the samefield, let M be a linear manifold of X , consider the quotient space X / M , and takethe natural quotient map π : X → X / M . If L ∈ L [ X , Z ] and if M ⊆ N ( L ) , thenthere exists a unique b L ∈ L [ X / M , Z ] such that L = b L ◦ π. In other words, the diagram X L −−−→ Z π (cid:31) ց x b L X / M commutes, which means the quotient map π factors the linear transformation L through X / M . Moreover, in this case N ( b L ) = N ( L ) / M and R ( b L ) = R ( L ) .Proof. See, e.g., [19, Theorem 3.4, 3.5] . (For a module version see, e.g., [15, TheoremV.4.7]; for a normed space version see, e.g., [16, Theorem 1.7.13]). (cid:3) Remark 2.1.
Let M be linear a manifold of a linear space X . A set { [e δ ] } δ ∈ ∆ ofcosets modulo M (with each e δ in X ) is a Hamel basis for the linear space X / M if and only if { e δ } δ ∈ ∆ is a Hamel basis for some algebraic complement of M (see,e.g., [12, Remark A.1(b)]) . Hence every Hamel basis { e γ } γ ∈ Γ for X is such that { [e γ ] } γ ∈ Γ includes a Hamel basis for X / M . Since the quotient map π : X → X / M is surjective, the image π ( E X ) of an arbitrary Hamel basis E X for X includes aHamel basis E X / M for X / M (i.e., E X / M ⊆ π ( E X )) . Thereforespan π ( E X ) = X / M for every Hamel basis E X for X , and so every element [ x ] of X / M can be written as a (finite) linear combination ofimages under π of elements of an arbitrary Hamel basis for X . C.S. KUBRUSLY
Bilinear Maps.
Let X , Y , Z be nonzero linear spaces over a field F . Take theCartesian product
X ×Y (no algebraic structure imposed on
X ×Y besides the factthat both X and Y are linear spaces) . A bilinear map φ : X ×Y → Z is a functionfrom the Cartesian product
X ×Y of linear spaces to a linear space Z whose sectionsare linear transformations. Precisely, let φ y = φ ( · , y ) = φ | X ×{ y } : X → Z be the y -section of φ and let φ x = φ ( x, · ) = φ | { x }×Y : Y → Z be the x -section of φ. Thesefunctions φ y and φ x are linear transformations: φ y = φ ( · , y ) ∈ L [ X , Z ] for each y in Y and φ x = φ ( x, · ) ∈ L [ Y , Z ] for each x in X . Let Z S denote the linear spaceover the same field F of all Z -valued functions on a set S , and let b [ X ×Y , Z ] = (cid:8) φ ∈ Z X ×Y : φ is bilinear (cid:9) stand for the collection of all Z -valued bilinear maps on X ×Y . The particular caseof Z = F yields a bilinear functional φ : X ×Y → F , also referred to as a bilinearform . Bilinearity of elements φ in b [ X ×Y , Z ] ensures b [ X ×Y , Z ] is a linear manifoldof the linear space Z X ×Y , thus a linear space over F itself . Let R ( φ ) = φ ( X ×Y )denote the range of φ ∈ b [ X ×Y , Z ], which in general is not a linear manifold of Z .As a composition of linear transformations is a linear transformation, a composi-tion of a bilinear map with a linear transformation is a bilinear map . Also, a restric-tion of a bilinear map to a Cartesian product of linear manifolds is again a bilinearmap (as a consequence of the definitions of linear manifold and of bilinear map) . Proposition 2.2.
Let X , Y , Z be linear spaces over the same field F , and let M and N be nonzero linear manifolds of X and Y . If φ : M×N → Z is a bilinearmap, then there exists a bilinear extension b φ : X ×Y → Z of φ defined on the wholeCartesian product X ×Y of the linear spaces X and Y .Sketch of Proof . Every linear transformation on a linear manifold of a linear spacehas a linear extension to the whole space, whose proof is an application of Zorn’sLemma (see, e.g., [10, Theorem 2.9]) . If a bilinear map is the product of two (alge-bra-valued) linear maps, then the proof is an application of the linear case . A prooffor the general bilinear case follows an argument similar to the linear case. (cid:3)
Bilinear maps can also be extended by factoring them by the natural bilinearmap through a tensor product space (see, e.g., [6, p.101]) . Indeed, Proposition 3.3will say that b [ X ×Y , Z ] ∼ = L [ X ⊗ Y , Z ], where X ⊗ Y stands for tensor product,and this ensures bilinear extension out of linear extension. It is, however, advisableto have the above extension result independently of the notion of tensor product.3.
Tensor Product of Linear Spaces: Axiomatic Theory
Definition 3.1.
Let X and Y be nonzero linear spaces over the same field F . A tensor product of X and Y is a pair ( T , θ ) consisting of a linear space T over F and a bilinear map θ : X ×Y → T fulfilling the following two axioms.(a) The bilinear map θ ∈ b [ X ×Y , T ] is such that its range R ( θ ) spans T .(b) If φ ∈ b [ X ×Y , Z ] is an arbitrary bilinear map into a linear space Z over F , thenthere is a linear transformation Φ ∈ L [ T , Z ] for which φ = Φ ◦ θ. That is, the diagram
ENSOR PRODUCTS REVISITED: AXIOMATIC APPROACH 5
X ×Y φ −−−→ Z θ (cid:31) ց x Φ T commutes, and so θ factors every bilinear map through T . The linear space T isreferred to as a tensor product space of X and Y associated with θ , and θ is referredto as the natural bilinear map (or simply the natural map ) associated with T .There are different interpretations of tensor products. For instance, the quotientspace and the linear maps of bilinear maps formulations are examples of commonprocedures for building tensor products . These will be shown to be isomorphic, andwill be exhibited in the next two sections . So the existence of tensor products willbe postponed until then . Definition 3.1 is our starting point for providing theseinterpretations . A similar start has been considered, for instance, in [14, SectionXVI.1] and [19, Chapter 14] for the quotient space formulation, in [24, Section 1.4]and [4, Chapter1] for both formulations, and in [7, Section 1.6] and [2, Section 2.2]for the linear maps of bilinear maps formulation aiming at tensor norms.The value θ ( x, y ) of the natural bilinear map θ : X ×Y → T = span θ ( X ×Y ) as-sociated with T at a pair ( x, y ) in X ×Y is denoted by x ⊗ y , x ⊗ y = θ ( x, y ) ∈ T for every ( x, y ) ∈ X ×Y , and x ⊗ y is called a single tensor (or a decomposable element , or a single tensorproduct of x and y ) in the tensor product space T . Take ( x, y ) ∈ X ×Y and α ∈ F arbitrary . Since θ is bilinear, then α ( x ⊗ y ) = ( αx ) ⊗ y = x ⊗ ( αy ) for any α ∈ F . So a multiple of a single tensor is again a single tensor, and the representationof a nonzero single tensor is not unique . Proofs for the next two propositions arestraightforward form Definition 3.1, thus omitted.
Proposition 3.1.
An element of a tensor product space is represented as a finitesum of single tensors ( and such a representation is not unique ) : ̥ ∈ T ⇐⇒ ̥ = X i x i ⊗ y i ( a finite sum ) . Proposition 3.2. If T is a tensor product space, then the linear transformation Φ ∈ L [ T , Z ] associated with each φ ∈ b [ X ×Y , Z ] as in Definition 3.1 is unique. The natural bilinear map θ associated with a tensor product space T is uniqueand, conversely, T associated with θ is unique . This is shown in the next theoremand its corollary . From now on all linear spaces anywhere are over the same field F . Theorem 3.1.
Let X and Y be linear spaces . Let ( T , θ ) and ( T ′ , θ ′ ) be tensorproducts of X and Y . Then there is a unique isomorphism Θ ∈ L [ T ′ , T ] such that Θ ◦ θ ′ = θ. That is, such that the following diagram commutes:
X ×Y θ −−−→ T θ ′ (cid:31) ց x Θ T ′ . Proof.
Let ( T , θ ) and ( T ′ , θ ′ ) be tensor products of X and Y . For any bilinear map φ : X ×Y → Z into a linear space Z there are linear transformations Φ : T → Z andΦ ′ : T ′ → Z such that φ = Φ ◦ θ = Φ ′ ◦ θ ′ (Definition 3.1), which means the diagrams C.S. KUBRUSLY
X ×Y φ −−−→ Z θ (cid:31) ց x Φ T and X ×Y φ −−−→ Z θ ′ (cid:31) ց x Φ ′ T ′ commute . Since θ ′ : X ×Y → T ′ and θ : X ×Y → T are bilinear maps, then there arelinear transformations Θ ′ : T → T ′ and Θ : T ′ → T such that θ ′ = Θ ′ ◦ θ and θ = Θ ◦ θ ′ (Definition 3.1 again), which means the diagrams X ×Y θ ′ −−−→ T ′ θ (cid:31) ց x Θ ′ T and X ×Y θ −−−→ T θ ′ (cid:31) ց x Θ T ′ commute . Therefore θ ′ = (Θ ′ ◦ Θ) ◦ θ ′ and θ = (Θ ◦ Θ ′ ) ◦ θ. Let I S denote the identity function on an arbitrary set S. By the above equations,Θ ′ ◦ Θ | R ( θ ′ ) = I R ( θ ′ ) : R ( θ ′ ) → R ( θ ′ ) and Θ ◦ Θ ′ | R ( θ ) = I R ( θ ) : R ( θ ) → R ( θ ) . Since Θ and Θ ′ are linear transformations (and so are their compositions), andsince span R ( θ ) = T and span R ( θ ′ ) = T ′ by Definition 3.1, then we getΘ ′ ◦ Θ = Θ ′ ◦ Θ | span R ( θ ′ ) = I span R ( θ ′ ) = I T ′ , Θ ◦ Θ ′ = Θ ◦ Θ ′ | span R ( θ ) = I span R ( θ ) = I T . Hence Θ and Θ ′ are the inverse of each other, and are unique by Proposition 3.2. (cid:3) Corollary 3.1.
A tensor product of linear spaces is unique up to an isomorphismin the following sense . If ( T , θ ) and ( T ′ , θ ′ ) are tensor products of the same pairof linear spaces, then ( T , θ ) = (Θ T ′ , Θ θ ′ ) for an isomorphism Θ in L [ T ′ , T ] . Inparticular, two tensor product spaces of the same pair of linear spaces coincide ifand only if the natural bilinear maps coincide.Proof.
This is an immediate consequence of Theorem 3.1. (cid:3)
A tensor product for a given pair of linear spaces is unique up to an isomorphismby Corollary 3.1 . Then for a given pair ( X , Y ) of linear space it is common to write T = X ⊗ Y for the tensor product space, and (
X ⊗ Y , θ ) for the tensor product, of X and Y . Proposition 3.3.
Take an arbitrary triple ( X , Y , Z ) of linear spaces . The linearspaces b [ X ×Y , Z ] and L [ T , Z ] are isomorphic: b [ X ×Y , Z ] ∼ = L [ X ⊗ Y , Z ] . Proof.
Take any Φ ∈ L [ T , Z ] . The composition Φ ◦ θ : X ×Y → Z lies in b [ X ×Y , Z ]since θ is bilinear and Φ is linear . Let C θ : L [ T , Z ] → b [ X ×Y , Z ] be defined by C θ (Φ) = Φ ◦ θ ∈ b [ X ×Y , Z ] for every Φ ∈ L [ T , Z ] . For every φ in b [ X ×Y , Z ] there is one and only one Φ in L [ T , Z ] for which φ = Φ ◦ θ according to Proposition 3.2 . Then C θ is injective and surjective, and so invertible . Since it is readily verified that C θ is linear, then C θ is an isomorphism. (cid:3) ENSOR PRODUCTS REVISITED: AXIOMATIC APPROACH 7
Thus a crucial property of a tensor product ( T , θ ) is to linearize bilinear maps(via factorization by θ through T ) in the sense of in Proposition 3.3 . In particular,(
X ⊗ Y ) ♯ ∼ = b [ X ×Y , F ] . Theorem 3.2.
Let T = X ⊗ Y be a tensor product space of X and Y , let E and D be nonempty subsets of X and Y respectively, and set ⊤ E,D = (cid:8) x ⊗ y ∈ X ⊗ Y : x ∈ E and y ∈ D (cid:9) . (a) If span E = X and span D = Y , then span ⊤ E,D = X ⊗ Y . (b) If E and D are linearly independent, then ⊤ E,D is linearly independent.Therefore, if E is a Hamel basis for X and D is a Hamel basis for Y , then ⊤ E,D is a Hamel basis for T = X ⊗ Y .Proof. (a) Take an arbitrary element ̥ = P ni =1 x i ⊗ y i from X ⊗ Y . If span E = X and span D = Y then consider any expansion of each x i ∈ X in terms of vectors e i,j from E and any expansion of each y i ∈ Y in terms of vectors d i,k from D. Since x ⊗ y = θ ( x, y ) for each pair ( x, y ) ∈ X ×Y , and since θ : X ×Y → X ⊗ Y is bilinear, ̥ = X ni =1 (cid:16)X mj =1 β j e i,j ⊗ X ℓk =1 γ k d i,k (cid:17) = X n,m,ℓi,j,k β j γ k ( e i,j ⊗ d i,k ) . Thus ̥ lies in span ⊤ E,D . Then
X ⊗ Y ⊆ span ⊤ E,D . So span ⊤ E,D = X ⊗ Y .(b) Let E ′ = { e j } mj =1 and D ′ = { d k } ℓk =1 be arbitrary nonempty finite linearly inde-pendent subsets of E and D respectively, and consider the linear manifolds M = span E ′ = span { e j } mj =1 ⊆ X and N = span D ′ = span { d k } ℓk =1 ⊆ Y . Set Z = L [ F m , F ℓ ], identified with the linear space of all ℓ × m matrices of entries in F . Take φ : M×N → Z given for u = P mj =1 β j e j ∈ M and v = P ℓk =1 γ k d k ∈ N by φ ( u, v ) = (cid:0) β j γ k (cid:1) ∈ Z for j = 1 , . . . , m and k = 1 , . . . , ℓ, where (cid:0) β j γ k (cid:1) is the ℓ × m matrix whose entries are the products β j γ k of the coeffi-cients of the unique expansion of arbitrary vectors u ∈ M and v ∈ N in terms of thelinearly independent sets E ′ and D ′ . It is readily verified that φ : M×N → Z is a bi-linear map . Thus consider the linear transformation Φ :
M ⊗ N → Z such that φ =Φ ◦ θ according to axiom (b) in Definition 3.1 . So φ ( u, v ) = Φ ( θ ( u, v )) = Φ( u ⊗ v )for every u = P mj =1 β j e j in M and every v = P ℓk =1 γ k d k in N . In particular,Φ( e j ⊗ d k ) = φ ( e j , d k ) = Π j,k , where Π j,k ∈ Z is the ℓ × m matrix whose entry at position j, k is 1 and all other en-tries are 0 . These matrices form a linearly independent set in Z . (In fact, { Π j,k } m,ℓj,k =1 is the canonical Hamel basis for Z .) Take any pair of integers k ′ , j ′ and suppose e j ′ ⊗ d k ′ is a linear combination of the remaining single tensors { e j ⊗ d k } j,k ∈ I ′ with I ′ = { j, k = 1 to m, ℓ : j = j ′ , k = k ′ } , say e j ′ ⊗ d k ′ = X j,k ∈ I ′ δ j,k ( e j ⊗ d k ) . Then, as Φ :
M ⊗ N → Z is linear,Π j ′ ,k ′ = Φ( e j ′ ⊗ d k ′ ) = X j,k ∈ I ′ δ j,k Φ( e j ⊗ d k ) = X j,k ∈ I ′ δ j,k Π j,k , C.S. KUBRUSLY and so δ j,k = 0 for every j, k ∈ I ′ since { Π j,k } m,ℓj,k =1 is linearly independent in Z . Hence { e j ⊗ d k } m,ℓj,k =1 is linearly independent in ⊤ E,D . In other words, ⊤ E ′ ,D ′ = { x ⊗ y ∈ X ⊗ Y : x ∈ E ′ and y ∈ D ′ } is a finite linearly independent subset of ⊤ E,D whenever E ′ and D ′ are finite linearlyindependent subsets of E and D. Thus if E and D are linearly independent subsetsof X and Y , then every finite subset of each of them is trivially linearly independent,and so is every finite subset of ⊤ E,D as we saw above . But if every finite subset of ⊤ E,D is linearly independent, then so is ⊤ E,D (see, e.g., [10, Proposition 2.3]). (cid:3)
Corollary 3.2. dim(
X ⊗ Y ) = dim
X · dim Y .Proof. If E and D are Hamel basis for X and Y , then ⊤ E,D is a Hamel basis for T byTheorem 3.2 . Also ⊤ E,D = (cid:8) x ⊗ y ∈ X ⊗ Y : x ∈ E and y ∈ D (cid:9) is in a one-to-onecorrespondence with E × D as E and D are linearly independent . Since E × D ) = E · D by definition of product of cardinal numbers (see, e.g., [10, Problem 1.30]),then ⊤ E,D = E · D. Thus follows the claimed dimension identity. (cid:3)
A straightforward consequence of the dimension identity of Corollary 3.2 is this . Tensor product is commutative up to an isomorphism:
X ⊗ Y ∼ = Y ⊗ X . Next we identify a special type of linear manifold of a tensor product space.
Proposition 3.4.
Let X and Y be linear spaces . Suppose M and N are linear man-ifolds of X and Y respectively . Let ( X ⊗ Y , θ ) be a tensor product . Set
M ⊗ N =span R ( θ | M×N ) . Then
M ⊗ N is a linear manifold of the tensor product space
X ⊗ Y and ( M ⊗ N , ϑ ) is a tensor product with ϑ = θ | M×N .Proof.
Consider Definition 3.1 . Let (
X ⊗ Y , θ ) be a tensor product . Take any biline-ar map φ : X ×Y → Z . Let Φ :
X ⊗ Y → Z be the linear transformation such that φ = Φ ◦ θ. Let M and N be linear manifolds of X and Y . Take the restriction θ | M×N of thenatural bilinear map θ to the Cartesian product M×N ⊆ X ×Y so that θ | M×N : M×N → R ( θ | M×N ) ⊆ span R ( θ | M×N ) ⊆ span R ( θ ) = X ⊗ Y , which is a bilinear map . Now consider the restriction φ | M×N of the arbitrary bi-linear map φ : X ×Y → Z to M×N , which is again a bilinear map for which φ | M×N = (Φ ◦ θ ) | M×N = Φ ◦ θ | M×N = Φ | span R ( θ | M×N ) ◦ θ | M×N , where Φ | span R ( θ | M×N ) is the restriction of the linear transformation Φ : X ⊗ Y → Z to the linear manifold span R ( θ | M×N ), again a linear transformation . Proposition2.2 says that every bilinear map ψ : M×N → Z is of the form φ | M×N : M×N → Z for some bilinear map φ : X ×Y → Z . Thus for every bilinear map ψ : M×N → Z there is a linear transformation Φ | span R ( θ | M×N ) : span R ( θ | M×N ) → Z such that φ = Φ | span R ( θ | M×N ) ◦ θ | M×N . Set
M⊗N = span R ( θ | M×N ) ⊆ X ⊗Y , a linear manifold of the linear space X ⊗ Y ,and ϑ = θ | M×N , the bilinear restriction of the bilinear θ. Thus by Definition 3.1(
M ⊗ N , ϑ ) is a tensor product . (cid:3) ENSOR PRODUCTS REVISITED: AXIOMATIC APPROACH 9
Therefore
M ⊗ N stands for the tensor product space of linear manifolds M and N of the linear spaces X and Y according to Corollary 3.1 and Proposition 3.4. Definition 3.2.
A linear manifold Υof a tensor product space T = X ⊗ Y is regular if Υ = M ⊗ N for some linear manifolds M and N of the linear spaces X and Y . Otherwise Υ is called irregular .The next characterization of regular linear manifolds is straightforward fromTheorem 3.2 . For a collection of properties of regular linear manifolds see [9, 11].
Proposition 3.5.
A nonzero linear manifold Υ of X ⊗ Y is regular if and only if
Υ = span ⊤ E ′ ,D ′ for some nonempty subsets E ′ ⊆ E and D ′ ⊆ D for Hamel bases E and D for X and Y respectively. The concept of tensor product of linear transformations is given as follows.
Definition 3.3.
Let X , Y , V , W be linear spaces and consider the tensor productspaces X ⊗ Y and
V ⊗ W . Let A ∈ L [ X , V ] and B ∈ L [ Y , W ] be linear transforma-tions . For each P ni =1 x i ⊗ y i in X ⊗ Y set( A ⊗ B ) P ni =1 x i ⊗ y i = P ni =1 Ax i ⊗ By i in V ⊗ W . This defines a map A ⊗ B of the linear space X ⊗ Y into the linear space
V ⊗ W ,which is referred to as the tensor product of the transformations A and B , or the tensor product transformation A ⊗ B . Proposition 3.6.
Take A ∈ L [ X , V ] and B ∈ L [ Y , W ] . (a) In fact, A ⊗ B in Definition 3.3 defines a linear transformation, A ⊗ B ∈ L [ X ⊗ Y , V ⊗ W ] , and ( A ⊗ B ) ̥ does not depend on the representation P ni =1 x i ⊗ y i of ̥ ∈ X ⊗ Y . (b) The map θ : L [ X , V ] ×L [ Y , W ] → L [ X ⊗ Y , V ⊗ W ] defined by θ ( A, B ) = A ⊗ B for every ( A, B ) ∈ L [ X , V ] ×L [ Y , W ] , with A ⊗ B ∈ L [ X ⊗ Y , V ⊗ W ] as in (a) , is bilinear . (c) Set L [ X , V ] ⊗ L [ Y , W ] = span R ( θ ) ⊆ L [ X ⊗ Y , V ⊗ W ] . Then ( L [ X , V ] ⊗ L [ Y , W ] , θ ) is a tensor product of L [ X , V ] and L [ Y , W ] . (d) The transformation A ⊗ B ∈ L [ X ⊗ Y , V ⊗ W ] in (a) coincides with a singletensor in the tensor product space L [ X , V ] ⊗ L [ Y , W ] : A ⊗ B ∈ L [ X , V ] ⊗ L [ Y , W ] ⊆ L [ X ⊗ Y , V ⊗ W ] . Proof.
Items (a), (b), (d) are readily verified . Item (c) goes as follows . Note thatspan R ( θ ) = span (cid:8) A ⊗ B ∈ L [ X ⊗ Y , V ⊗ W ] : A ∈ L [ X , V ] B ∈ L [ Y , W ] (cid:9) = (cid:8)P ni =1 A i ⊗ B i ∈ L [ X ⊗ Y , V ⊗ W ] : A ∈ L [ X , V ] , B ∈ L [ Y , W ] , n ∈ N ] (cid:9) . For each bilinear map φ : L [ X , V ] ×L [ Y , W ] → Z take Φ : span R ( θ ) → Z defined byΦ (cid:16)X ni =1 A i ⊗ B i (cid:17) = X ni =1 φ ( A i , B i ) ∈ Z for every X ni =1 A i ⊗ B i ∈ span R ( θ ) . Since φ is bilinear, it is easy to show that Φ is linear: Φ ∈ L [span R ( θ ) , Z ] . Moreover,for every (
A, B ) ∈ L [ X , V ] ×L [ Y , W ](Φ ◦ θ )( A, B ) = Φ (cid:0) θ ( A, B ) (cid:1) = Φ( A ⊗ B ) = φ ( A, B ) . Thus φ = Φ ◦ θ , equivalently, the diagram L [ X , V ] ×L [ Y , W ] φ −−−→ Z θ (cid:31) ց x Φ span R ( θ )commutes. Therefore (span R ( θ ) , θ ) satisfies the axioms of Definition 3.1. (cid:3) The particular case of V = W = F is worth noticing . In this case L [ X , V ] = L [ X , F ] = X ♯ , L [ Y , W ] = L [ Y , F ] = Y ♯ and L [ X ⊗ Y , V ⊗ W ] = L [ X ⊗ Y , F ⊗ F ] = L [ X ⊗ Y , F ] = ( X ⊗ Y ) ♯ Indeed, dim( F ⊗ F ) = dim F = 1 by Corollary 3.2. So write F ⊗ F = F for F ⊗ F ∼ = F as usual . Since L [ X , V ] ⊗ L [ Y , W ] ⊆ L [ X ⊗ Y , V ⊗ W ] by Proposition 3.6(c), then X ♯ ⊗ Y ♯ ⊆ ( X ⊗ Y ) ♯ . Basic results on tensor product transformations are given next . Most are straight-forward or readily verified: properties (a,b) are trivial since A ⊗ B is a single tensor,(c,d) are straightforward by definition of A ⊗ B , and (e,f) are readily verified forthe regular linear manifolds N ( A ) ⊗ N ( B ) and R ( A ) ⊗ R ( B ) . For the nonreversibleinclusion in (f) see, e.g., [13] . Item (g) says: tensor product of linear transforma-tions is commutative up to isomorphisms, which means A ⊗ B and B ⊗ A are iso-morphically equivalent in the sense that Π ( A ⊗ B ) = ( B ⊗ A )Π for isomorphismsΠ : X ⊗ Y → Y ⊗ X and Π : V ⊗ W → W ⊗ V (whose existence follows from thefact that
X ⊗ Y ∼ = Y ⊗ X as a consequence of Corollary 3.2) . We prove (h) below.
Proposition 3.7.
Let V , W , X , Y , X ′ , Y ′ be linear spaces . Take
A, A , A ∈ L [ X , V ] B, B , B ∈ L [ Y , W ] , C ∈ L [ X ′ , X ] , D ∈ L [ Y ′ , Y ] and also α, β ∈ F . Then (a) α β ( A ⊗ B ) = αA ⊗ βB = α βA ⊗ B = A ⊗ α βB , (b) ( A + A ) ⊗ ( B + B ) = A ⊗ B + A ⊗ B + A ⊗ B + A ⊗ B , (c) AC ⊗ BD = ( A ⊗ B ) ( C ⊗ D ) , (d) If A and B are invertible, then so is A ⊗ B and ( A ⊗ B ) − = A − ⊗ B − , (e) R ( A ) ⊗ R ( B ) = R ( A ⊗ B ) , (f) N ( A ) ⊗ N ( B ) $ N ( A ⊗ B ) , (g) A ⊗ B ∼ = B ⊗ A . (h) ( A ⊗ B ) ♯ = A ♯ ⊗ B ♯ .Proof. (h) Take A ∈ L [ X , V ], B ∈ L [ Y , W ], A ♯ ∈ L [ V ♯ , X ♯ ], B ♯ ∈ L [ W ♯ , Y ♯ ] . Considerthe single tensors A ⊗ B in L [ X , V ] ⊗ L [ Y , W ] ⊆ L [ X ⊗ Y , V ⊗ W ] and A ♯ ⊗ B ♯ in L [ V ♯ , X ♯ ] ⊗ L [ W ♯ , Y ♯ ] ⊆ L [ V ♯ ⊗ W ♯ , X ♯ ⊗ Y ♯ ], and the algebraic adjoint ( A ⊗ B ) ♯ in L [( Y ⊗ W ) ♯ , ( X ⊗ V ) ♯ ] . Take an arbitrary f ⊗ g ∈ V ♯ ⊗ W ♯ ⊆ ( V ⊗ Y ) ♯ . By defi-nition of tensor product transformation and of algebraic adjoint( A ♯ ⊗ B ♯ )( f ⊗ g ) = A ♯ f ⊗ B ♯ g = fA ⊗ gB ∈ X ♯ ⊗ Y ♯ ⊆ ( X ⊗ Y ) ♯ , ( ∗ ) ENSOR PRODUCTS REVISITED: AXIOMATIC APPROACH 11 ( A ⊗ B ) ♯ ( f ⊗ g ) = ( f ⊗ g )( A ⊗ B ) ∈ ( X ⊗ Y ) ♯ . ( ∗∗ ) Claim . ( f ⊗ g )( A ⊗ B ) = fA ⊗ gB ∈ X ♯ ⊗ Y ♯ ⊆ ( X ⊗ Y ) ♯ . Proof . Take an arbitrary single tensor x ⊗ y in X ⊗ Y , and an arbitrary single ten-sor A ⊗ B in L [ X , V ] ⊗ L [ Y , W ] . By definition of tensor product transformation,( A ⊗ B )( x ⊗ y ) = Ax ⊗ By , a single tensor in V ⊗ W . Next take an arbitrary singletensor f ⊗ g in V ♯ ⊗ W ♯ = L [ V , F ] ⊗ L [ W , F ] so that, by definition of tensor producttransformation, ( f ⊗ g )( Ax ⊗ By ) = f Ax ⊗ gBy ∈ F ⊗ F = F . On the other hand,since f A ∈ X ♯ = L [ X , F ] and gB ∈ Y ♯ = L [ Y , F ], then (definition of tensor producttransformation), ( f A ⊗ gB )( x ⊗ y ) = f Ax ⊗ gBy ∈ F ⊗ F = F . Summing up:( f ⊗ g )( A ⊗ B )( x ⊗ y ) = ( f ⊗ g )( Ax ⊗ By ) = fAx ⊗ gBy = ( fA ⊗ gB )( x ⊗ y )for every x ⊗ y ∈ X ⊗ Y . Thus ( f ⊗ g )( A ⊗ B ) ̥ = ( f ⊗ g )( A ⊗ B ) P i x i ⊗ y i = P i ( f ⊗ g )( A ⊗ B )( x i ⊗ y i ) = P i ( fA ⊗ gB )( x i ⊗ y i ) = ( fA ⊗ gB ) P i x i ⊗ y i =( fA ⊗ gB ) ̥ for every ̥ ∈ X ⊗ Y , and hence( f ⊗ g )( A ⊗ B ) = fA ⊗ gB in X ♯ ⊗ Y ♯ = L [ X , F ] ⊗ L [ Y , F ] ⊆ L [ X ⊗ Y , F ] = ( X ⊗ Y ) ♯ . (cid:3) Then by ( ∗ ), ( ∗∗ ) and the above claim( A ♯ ⊗ B ♯ )( f ⊗ g ) = ( A ⊗ B ) ♯ ( f ⊗ g )for every single tensor f ⊗ g ∈ X ♯ ⊗ Y ♯ . Thus by a similar argument A ♯ ⊗ B ♯ = ( A ⊗ B ) ♯ in L [ X ♯ ⊗ Y ♯ , F ] = ( X ♯ ⊗ Y ♯ ) ♯ , concluding the proof of (g). (cid:3) An Interpretation via Quotient Space
Let X and Y be nonzero linear spaces over a field F . Take the Cartesian product S = X ×Y of X and Y . Consider the notation and terminology of Subsection 2.1 . Thus S / is the free linear space generated by S (i.e., the linear space of all functions f : X ×Y → F that vanish everywhere on the complement of some finite subset of X ×Y ), and S = { e ( x,y ) } ( x,y ) ∈ S is the Hamel basis for S / consisting of characteristicfunctions e ( x,y ) = χ { ( x,y ) } = χ { x } χ { y } : X ×Y → F of all singletons at each pairof vectors ( x, y ) in the Cartesian product S = X ×Y . With the identification ≈ ofSubsection 2.1 still in force, take the sums of elements from S whose double indiceshave one common entry and consider the following differences . (i) e ( x + x ,y ) − e ( x ,y ) − e ( x ,y ) ≈ ( x + x , y ) − ( x , y ) − ( x , y ),(ii) e ( x,y + x ) − e ( x,y ) − e ( x,y ) ≈ ( x , y + y ) − ( x , y ) − ( x , y ),(iii) e ( αx,y ) − α e ( x,y ) ≈ ( αx, y ) − α ( x, y ),(iv) e ( x,αy ) − α e ( x,y ) ≈ ( x, αy ) − α ( x, y ),for every x, x , x ∈ X , every y, y , y ∈ Y , and every α ∈ F . The above differencesare not null . If they were, then we could identify a bilinear rule on the ordered pairs( x, y ) ∈ X ×Y . Thus look at equivalence classes [ e ( x,y ) ] of characteristic functions e ( x,y ) in S ⊆ S / , gathering those differences at the origin of a quotient space as fol-lows . Take the linear manifold M of S / generated by the differences in (i)–(iv), viz., M = span (cid:8) e ( x + x ,y ) − e ( x ,y ) − e ( x ,y ) , e ( x,y + x ) − e ( x,y ) − e ( x,y ) ,e ( αx,y ) − α e ( x,y ) , e ( x,αy ) − α e ( x,y ) (cid:9) . Take the quotient space S // M of S / modulo M , consider the natural quotient map π : S / → S // M , and define a map θ : X ×Y → S // M on S = X ×Y as follows . For every ( x, y ) ∈ X ×Y set θ ( x, y ) = π ( e ( x,y ) ) . Therefore θ ( S ) = π ( S ) . The Hamel basis S = { e ( x,y ) } ( x,y ) ∈ S for the linear space S / generated by S is naturallyidentified with S itself, and so the domain of θ is identified with the domain of therestriction π | S of the natural quotient map π to S , that is, S ≈ S . Since they alsocoincide pointwise, then θ is naturally identified with π | S . So write θ = π | S for θ ≈ π | S . Elements of S // M which are images of the map θ : X ×Y → S // M are denoted by x ⊗ y , and again referred to as single tensors or decomposable elements : x ⊗ y = θ ( x, y ) = π ( e ( x,y ) ) = [ e ( x,y ) ] for every ( x, y ) ∈ X ×Y . Theorem 4.1. ( S // M , θ ) is a tensor product of X and Y .Proof. Consider the axioms (a) and (b) in Definition 3.1.(a ) By definition of M , the differences in (i) to (iv) lie in M = [0] ∈ S // M . Thuswith θ ( x, y ) = π ( e ( x,y ) ) = [ e ( x,y ) ] it follows that θ : X ×Y → S // M is a bilinear map.(a ) Since S = { e ( x,y ) } ( x,y ) ∈X ×Y is a Hamel basis for the linear space S / , thenspan π ( S ) = S // M (cf . Remark 2.1). Thus as R ( θ ) = R ( π | S ),span R ( θ ) = span R ( π | S ) = span π ( S ) = S // M . (b) Take a bilinear map φ : X ×Y → Z into a linear space Z and consider a trans-formation e Φ on the free linear space S / generated by the Cartesian product X ×Y , e Φ : S / → Z , defined by e Φ( f ) = X ni =1 α i φ ( x i , y i ) ∈ Z for every f = X ni =1 α i e ( x i ,y i ) ∈ S /, which is clearly linear . Moreover, since e Φ( e ( x i ,y i ) ) = φ ( x, y ), then e Φ( S ) = φ ( S ) . Again, since S ≈ S = X ×Y , then e Φ | S is naturally identified with φ. So write e Φ | S = φ. Furthermore, since φ : X ×Y → Z is bilinear, then e Φ evaluated at the differences in(i) to (iv) is null . Hence the linear manifold M of S / is such that e Φ( M ) = 0 . Thatis,
M ⊆ N ( e Φ) . Thus by Proposition 2.1 there exists a unique linear transformationΦ : S // M → Z such that e Φ = Φ ◦ π. Therefore restricting to S ⊂ S / and since span π ( S ) = S // M ,we get φ = e Φ | S = (Φ ◦ π ) | S = Φ | span R ( π ) ◦ π | S = Φ ◦ θ. Equivalently, the diagrams
ENSOR PRODUCTS REVISITED: AXIOMATIC APPROACH 13 S / e Φ −−−→ Z π (cid:31) ց x Φ S // M and S φ −−−→ Z θ (cid:31) ց x Φ S // M commute . Identifying again S ≈ S (thus regarding S = X ×Y as a subset of S / andwriting θ = π | S for θ = π | S and φ = e Φ | S for φ = e Φ | S ), then the diagram X ×Y φ −−−→ Z θ (cid:31) ց x Φ S // M . commutes . Therefore the pair ( S // M , θ ) satisfies the axioms of Definition 3.1. (cid:3) An Interpretation via Linear Map of Bilinear Maps
Let X and Y be nonzero linear spaces over the same field F , take the Cartesianproduct X ×Y of X and Y , and consider the linear space b [ X ×Y , F ] of all bilinearmaps into an arbitrary but fixed linear space F over F . Associated with each pair( x, y ) ∈ X ×Y , consider a transformation x ⊗ y : b [ X ×Y , F ] → F defined by( x ⊗ y )( ψ ) = ψ ( x, y ) ∈ F for every ψ ∈ b [ X ×Y , F ] . This again is referred to as a single tensor and as is readily verified x ⊗ y is a lineartransformation on the linear space of bilinear maps, x ⊗ y ∈ L [ b [ X ×Y , F ] , F ] . Thus the term linear maps of bilinear maps means that this approach to tensorproduct focuses explicitly on the linearization of bilinear maps . Take the collection ⊤ X , Y , F = (cid:8) x ⊗ y ∈ L [ b [ X ×Y , F ] , F ] : x ∈ X and y ∈ Y (cid:9) of all single tensors . Consider its span, span ⊤ X , Y , F ⊆ L [ b [ X ×Y , F ] , F ], which is alinear manifold of the linear space L [ b [ X ×Y , F ] , F ], and define a map θ : X ×Y → ⊤ X , Y , F ⊆ span ⊤ X , Y , F as follows: for each pair ( x, y ) ∈ X ×Y set θ ( x, y ) = x ⊗ y. Theorem 5.1. (span ⊤ X , Y , F , θ ) is a tensor product of X and Y .Proof. Take an arbitrary linear space F . Consider axioms (a), (b) in Definition 3.1.(a ) θ ( x, y ) is a linear transformation, θ ( x, y ) ∈ L [ b [ X ×Y , F ] , F ] for each ( x, y ) in X ×Y , which is given by θ ( x, y )( ψ ) = ψ ( x, y ) ∈ F for every ψ ∈ b [ X ×Y , F ] . Thenbilinearity of ψ is transferred to θ. Hence θ ∈ b [ X ×Y , span ⊤ X , Y , F ].(a ) Since θ ( x, y ) = x ⊗ y , then R ( θ ) = ⊤ X , Y , F , and so span R ( θ ) = span ⊤ X , Y , F .(b) An arbitrary element ̥ of the linear space span ⊤ X , Y , F is a linear combinationof single tensors, thus lying in L [ b [ X ×Y , F ] , F ] . Since θ is bilinear, then every ̥ in span ⊤ X , Y , F is a finite sum of single tensors x ⊗ y = θ ( x, y ): ̥ = X i x i ⊗ y i ∈ span ⊤ X , Y , F ⊆ L [ b [ X ×Y , F ] , F ] . Given an arbitrary bilinear map φ ∈ b [ X ×Y , Z ] into any linear space Z , considerthe transformation Φ : span ⊤ X , Y , F → Z defined byΦ( ̥ ) = X i φ ( x i , y i ) ∈ Z for every ̥ = X i x i ⊗ y i ∈ span ⊤ X , Y , F . As is readily verified, this is a linear transformation: Φ ∈ L [ span ⊤ X , Y , F , Z ] . Also,(Φ ◦ θ )( x, y ) = Φ (cid:0) θ ( x, y ) (cid:1) = Φ( x ⊗ y ) = φ ( x, y )for every ( x, y ) ∈ X ×Y . Hence φ = Φ ◦ θ , leading to the commutative diagram X ×Y φ −−−→ Z θ (cid:31) ց x Φ span ⊤ X , Y , F and therefore the pair (span ⊤ X , Y , F , θ ) satisfies the axioms of Definition 3.1. (cid:3) This interpretation, where single tensors are defined as linear transformationsof bilinear maps, is specially tailored to highlight the central property of tensorproducts as a tool to linearize bilinear maps according to Proposition 3.3 . An important particular case refers to linear and bilinear forms by setting F = F . In this case single tensors are linear forms of bilinear forms, x ⊗ y ∈ ⊤ X , Y , F ⊆ L [ b [ X ×Y , F ] , F ] = b [ X ×Y , F ] ♯ , in the algebraic dual of b [ X ×Y , F ] . Particularizing still further, besides setting F = F , replace the above linear space b [ X ×Y , F ] by the following subset of it: b X ♯ ×Y ♯ [ X ×Y , F ] = (cid:8) ψ ∈ b [ X ×Y , F ] : ψ ( x, y ) = µ ( x ) ν ( y ) for ( µ, ν ) ∈ X ♯ ×Y ♯ (cid:9) , consisting of products of linear forms µ ∈ X ♯ = L [ X , F ] and ν ∈ Y ♯ = L [ Y , F ] . Theset b X ♯ ×Y ♯ [ X ×Y , F ] is not a linear manifold of b [ X ×Y , F ] . Define single tensorsas before . To each ( x, y ) ∈ X ×Y associate a function x ⊗ y : b X ♯ ×Y ♯ [ X ×Y , F ] → F defined for every ψ ∈ b X ♯ ×Y ♯ [ X ×Y , F ] by( x ⊗ y )( µ, ν ) = ( x ⊗ y )( ψ ) = ψ ( x, y ) = µ ( x ) ν ( y ) for every ( µ, ν ) ∈ X ♯ ×Y ♯ . The difference between this and the previous procedure is due to the fact that singletensors are not linear transformations (or linear forms) any longer as their domain b X ♯ ×Y ♯ [ X ×Y , F ] is not a linear space . However, as is readily verified they can beregarded as bilinear forms on the Cartesian product of the linear spaces X ♯ and Y ♯ : x ⊗ y ∈ b [ X ♯ ×Y ♯ , F ] = b [ L [ X , F ] ×L [ Y , F ] , F ] . Thus take the collection ⊤ ′X , Y of all these single tensors ⊤ ′X , Y = (cid:8) x ⊗ y ∈ b [ X ♯ ×Y ♯ , F ] : x ∈ X and y ∈ Y (cid:9) , and consider its span, span ⊤ ′X , Y , which is now a linear manifold of the linear space b [ X ♯ ×Y ♯ , F ] of all bilinear forms of pairs of linear forms. As before, define a map θ ′ : X ×Y → ⊤ ′X , Y ⊆ span ⊤ ′X , Y for each pair ( x, y ) ∈ X ×Y by θ ′ ( x, y ) = x ⊗ y. Now the value of θ ′ at ( x, y ) ∈ X ×Y is a bilinear form, θ ′ ( x, y ) ∈ b [ X ♯ ×Y ♯ , F ], whichis given by θ ′ ( x, y )( µ, ν ) = µ ( x ) ν ( y ) ∈ F for every ( µ, ν ) ∈ X ♯ ×Y ♯ . Corollary 5.1. (span ⊤ ′X , Y , θ ′ ) is a tensor product of X and Y . ENSOR PRODUCTS REVISITED: AXIOMATIC APPROACH 15
Proof.
Consider the proof of Theorem 5.1 . Replace F by F so that L [ b [ X ×Y , F ] , F ]is replaced by L [ b [ X ×Y , F ] , F ] . Then replace the linear space b [ X ×Y , F ] by thesubset b X ♯ ×Y ♯ [ X ×Y , F ], still keeping the same definition of single tensors, so that ⊤ X , Y , F ⊆ L [ b [ X ×Y , F ] , F ] is replaced by ⊤ ′X , Y ⊆ b [ L [ X , F ] ×L [ Y , F ] , F ] . Again, θ ′ : X ×Y → span ⊤ ′X , Y , F is a bilinear map with span R ( θ ′ ) = ⊤ ′X , Y , F . Thus the ar-gument in the proof of Theorem 5.1 still holds, associating with each bilinear map φ : X ×Y → Z the same linear transformation Φ into Z now acting on span ⊤ ′X , Y . (cid:3) Remark 5.1.
Here is a common and useful example of such a particular case (with F = C ) . Let X and Y be complex Hilbert spaces with inner products h · ; · i X and h · ; · i Y , which are sesquilinear forms (not bilinear forms) . Algebraic duals X ♯ and Y ♯ are now naturally replaced by topological duals X ∗ and Y ∗ of continuous linearfunctionals . A single tensor for each ( x, y ) ∈ X ×Y is usually defined in this case by( x ⊗ y )( u, v ) = h x ; u i X h y ; v i Y for every ( u, v ) ∈ X ×Y . (See e.g., [18, Section II.4] and [8].) The Riesz Representation Theorem for Hilbertspaces says that µ lies in X ∗ and ν lies in Y ∗ if and only if µ ( · ) = h · ; u i X and ν ( · ) = h · ; v i Y for some u in X and v in Y . Thus identify µ ∈ X ∗ and ν ∈ Y ∗ with u ∈ X and v ∈ Y such that the pair ( u, v ) ∈ X ×Y is identified with the pair ( µ, ν ) ∈ X ∗ ×Y ∗ . Then a single tensor x ⊗ y associated with a pair ( x, y ) ∈ X ×Y is in fact a bilinearform x ⊗ y : X ∗ ×Y ∗ → C in b [ X ∗ ×Y ∗ , C ], which is equivalently written as( x ⊗ y )( µ, ν ) = µ ( x ) ν ( y ) for every ( µ, ν ) ∈ X ∗ ×Y ∗ . Final Remarks
Multiple Tensor Products.
It is clear how the preceding arguments (in Sec-tions 3, 4 and 5) can be naturally extended to cover the notion of an algebraic tensorproduct of a finite collection {X i } ni =1 of linear spaces over the same field, yielding atensor product space N ni =1 X i of a finite number of linear spaces . This is based onthe notions of multiple Cartesian products Q ni =1 X i , n-tuples, and multilinear mapsas a natural extension of Cartesian product of two linear spaces, ordered pairs, andbilinear maps . All results in Sections 3, 4 and 5 remain true (essentially with thesame statement, following similar arguments) if extended to such multiple tensorproducts . To extend a result on tensor product from a pair of linear spaces to an n -tuple (or to an ∞ -tuple) may be a relevant task . Sometimes this is a simple job(achieved by induction) but not always . On the other hand, what may also notbe always simple is the other way round: when a notion is initially defined for an n -tuple, it may be wise to go down to a pair to see clearly what is really going on.6.2. Tensor Products of Banach Spaces.
We have been dealing with algebraictensor products
X ⊗ Y of linear spaces X and Y . A natural follow-up is to equip theunderlying linear space
X ⊗ Y with a norm and advance the theory of tensor prod-ucts to Banach spaces . So a new starting point is to equip
X ⊗ Y with a suitablenorm . If X and Y are Banach spaces and X ∗ and Y ∗ are their duals, then let x ⊗ y and f ⊗ g be single tensors in the tensor product spaces X ⊗ Y and X ∗ ⊗ Y ∗ . A norm k · k on X ⊗ Y is a reasonable crossnorm if, for every x ∈ X , y ∈ Y , f ∈ X ∗ , g ∈ X ∗ ,(a) k x ⊗ y k ≤ k x k k y k ,(b) f ⊗ g lies in ( X ⊗ Y ) ∗ , and k f ⊗ g k ∗ ≤ k f k k g k (where k · k ∗ is the normon the dual ( X ⊗ Y ) ∗ when ( X ⊗ Y ) is equipped with the norm in (a)), so that X ∗ ⊗ Y ∗ ⊆ ( X ⊗ Y ) ∗ . It can be verified that (i) the above norm inequalitiesbecome identities, and (ii) when restricted to X ∗ ⊗ Y ∗ the norm k · k ∗ on ( X ⊗ Y ) ∗ is again a reasonable crossnorm (with respect to ( X ∗ ⊗ Y ∗ ) ∗ ) . Two special normson
X ⊗ Y are the so-called injective k · k ∨ and projective k · k ∧ norms, k ̥ k ∨ = sup k f k≤ , k g k≤ , f ∈X ∗ , g ∈Y∗ (cid:12)(cid:12)(cid:12)X i f ( x i ) g ( y i ) (cid:12)(cid:12)(cid:12) , k ̥ k ∧ = inf { x i } i , { y i } i , ̥ = P i x i ⊗ y i X i k x i k k y i k , for every ̥ = P i x i ⊗ y i ∈ X ⊗ Y . It can be shown that (iii) these are reasonablecrossnorms, and (iv) a norm k · k on X ⊗ Y is a reasonable crossnorm if and only if k ̥ k ∨ ≤ k ̥ k ≤ k ̥ k ∧ for every ̥ ∈ X ⊗ Y . Anyhow, equipped with any reasonable crossnorm, a tensor product space
X ⊗ Y of a pair of Banach spaces X and Y is not necessarily complete . Thus one takes thecompletion X b ⊗Y of X ⊗ Y . For the theory of the Banach space X b ⊗Y the readeris referred, for instance, to [7, 2, 20, 3] . If X and Y are Hilbert spaces, then X b ⊗Y becomes a Hilbert space when one takes the reasonable crossnorm on X ⊗ Y thatnaturally comes from the inner products in X and Y as in Remark 5.1, by setting h x ⊗ y , x ⊗ y i X ⊗Y = h x ⊗ x i X h y ⊗ y i Y (see, e.g., [18, 22, 8]). Acknowledgment
The author thanks the anonymous referees for their constructive criticisms.
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