5.4 Tensor Product Space Structure
The tensor product space structure merges vector spaces into a new space, enabling multilinear operations and forming the foundation of tensor algebra.
Tensor Product Space Structure is the layered description of V ⊗ W as an algebraic object, built up from a carrier set of cosets, an addition operation on that carrier set, and a scalar action of the base field on it, together satisfying the vector space axioms and thereby justifying the space's role as the vector space in which multilinear maps are identified with elements.
The Layers, in Order
Carrier Set First
The carrier structure fixes the underlying set of V ⊗ W: the cosets x + R of the free vector space F(V × W) modulo the relation subspace R. No operation is specified at this layer; it names only which elements exist, established as a direct consequence of the earlier quotient formation stage of the construction.
Addition Layered on the Carrier
The addition structure defines (x + R) + (y + R) = (x + y) + R, giving the carrier set the structure of an abelian group and, crucially, supplying the mechanism by which decomposable elements combine into non-decomposable ones — a fact with no analogue at the carrier-set level alone, since the carrier set by itself says nothing about how its elements relate to one another algebraically.
Scalar Action Layered on Top
The scalar action structure defines c(x + R) = (cx) + R, distributing over the addition already in place and satisfying the field axioms relating scalar multiplication to addition; it is this layer that reconciles scaling a decomposable element with scaling either of its individual factors, via c(v ⊗ w) = (cv) ⊗ w = v ⊗ (cw).
What the Combined Layers Establish
A Genuine Vector Space Over F
Taken together, the carrier set, its addition, and its scalar action satisfy every vector space axiom over F: closure, associativity and commutativity of addition, existence of an additive identity and inverses, distributivity of scalar multiplication over vector addition and over field addition, compatibility of scalar multiplication with field multiplication, and the identity scalar acting trivially. None of these axioms is verified from first principles at the tensor product level; each is inherited from the corresponding axiom already holding in the free vector space F(V × W), since quotienting by a subspace preserves every vector space identity that held before the quotient.
The Structure Presupposed by Every Later Result
Every subsequent structural fact about V ⊗ W — its dimension, its induced basis, the behavior of induced maps f ⊗ g upon it — is stated relative to this three-layer vector space structure already being in place; dimension and basis, for instance, are meaningless without an addition and scalar action already fixed to make linear independence and spanning well-defined notions.
Relation to Basis Dependence and the Space's Role as a Vector Space
Structure Independent of Any Chosen Basis
The carrier set, addition, and scalar action are all defined without reference to any basis of V or W; a basis is introduced only afterward, to give coordinates to elements already existing independently of that choice, which is the subject taken up separately under basis dependence.
Structure That Licenses Treating V ⊗ W as an Ordinary Vector Space
Because the three layers together satisfy the full set of vector space axioms, V ⊗ W may be treated, in every subsequent argument, exactly as any other vector space would be — subject to linear maps, spanned by bases, decomposed into subspaces — which is the content taken up separately under the tensor product space's vector space role.