5 Tensor Product Theory

Tensor Product Theory is the branch of tensor algebra concerned with the construction of the tensor product V ⊗ W of two vector spaces, the universal property that characterizes it uniquely, and the algebraic structure of the resulting space, treated as a foundation on which multilinear maps, tensor types, and the graded tensor algebra are all subsequently built.

The Central Construction

From Bilinear Maps to a Single Linear Object

The motivating problem of tensor product theory is that bilinear maps out of V × W are common and useful but do not themselves form a vector space closed under composition with linear maps in a convenient way; the tensor product resolves this by producing a single vector space V ⊗ W together with a fixed bilinear map

such that every bilinear map out of V × W factors uniquely as a linear map out of V ⊗ W composed with this fixed map. Constructing V ⊗ W reduces the entire theory of bilinear maps on V × W to the theory of linear maps on a single space.

Explicit Construction as a Quotient

Concretely, V ⊗ W is built as the quotient of the free vector space generated by all symbols (v, w) for v in V and w in W, by the subspace generated by the relations

and the analogous relation in the second argument, along with the relations enforcing compatibility with scalar multiplication. The image of (v, w) in the quotient is written v ⊗ w, and the relations imposed are exactly what forces the map (v, w) ↦ v ⊗ w to be bilinear.

The Universal Property

Statement

For every vector space U and every bilinear map B: V × W → U, there exists a unique linear map B̃: V ⊗ W → U satisfying B̃(v ⊗ w) = B(v, w) for all v, w. This property, rather than the specific quotient construction used to obtain it, is what characterizes V ⊗ W up to unique isomorphism: any two vector spaces satisfying the same universal property are canonically isomorphic, so the tensor product is independent of the particular construction chosen to build it.

Consequences

The universal property immediately gives functoriality — a pair of linear maps f: V → V' and g: W → W' induces a unique linear map f ⊗ g: V ⊗ W → V' ⊗ W' sending v ⊗ w to f(v) ⊗ g(w) — and it gives the identification of multilinear maps with linear maps out of iterated tensor products, extending the two-factor case to any finite number of factors.

Structural Results

Dimension and Basis

When V and W are finite-dimensional with bases {e_i} and {f_j}, the elements e_i ⊗ f_j form a basis of V ⊗ W, so

and every element of V ⊗ W is a finite sum of decomposable elements v ⊗ w, though such a sum need not itself be decomposable — the existence of non-decomposable, or entangled, elements once more than one basis term is combined is one of the theory's central structural facts.

Iteration and Associativity

The tensor product extends to any finite number of vector spaces, and (V ⊗ W) ⊗ U is canonically isomorphic to V ⊗ (W ⊗ U), so parentheses may be dropped and V_1 ⊗ V_2 ⊗ ⋯ ⊗ V_k is written without ambiguity; this associativity is what allows the tensor product to be iterated into the graded tensor algebra without needing to fix an order of operations in advance.

Scope Within Tensor Algebra

Tensor product theory supplies the construction, universal property, and internal vector space structure on which the classification of multilinear maps, the graded tensor algebra, and specific named tensor types all depend, functioning as the shared foundation those later topics reference rather than re-derive.