1.2.28 Tensor Product Definition

The tensor product combines vector spaces into a new space, representing multilinear relationships and forming the basis of tensor algebra.

Tensor Product Definition is the characterization of a construction that combines two vector spaces into a new vector space in a way that converts bilinear maps defined on the original spaces into linear maps defined on the new space. The tensor product is the fundamental operation of tensor algebra: it is the mechanism through which tensors of any order are built up from vectors and covectors, and it provides the universal setting in which multilinear phenomena become linear.

Formal Definition via Universal Property

Let $V$ and $W$ be vector spaces over a field $F$ . The tensor product of $V$ and $W$ is a vector space $V \otimes W$ , together with a bilinear map

\otimes : V \times W \to V \otimes W

sending a pair $(v, w)$ to an element denoted $v \otimes w$ , satisfying the following universal property: for every vector space $U$ and every bilinear map

B : V \times W \to U

there exists a unique linear map

\tilde{B} : V \otimes W \to U

such that $B (v, w) = \tilde{B} (v \otimes w)$ for every $v \in V$ and $w \in W$ . This factorization property is called the universal property of the tensor product, and it characterizes $V \otimes W$ uniquely up to a canonical isomorphism: any two vector spaces satisfying this property with respect to the same bilinear map are canonically identified with one another.

Explicit Construction

Free Vector Space Modulo Bilinearity Relations

The tensor product can be constructed explicitly as a quotient of a free vector space. Let $F (V \times W)$ denote the free vector space generated by all ordered pairs $(v, w)$ with $v \in V$ and $w \in W$ , meaning every such pair is treated as an independent basis vector. Let $N$ be the subspace generated by all elements of the form

(a v_{1} + b v_{2}, w) - a (v_{1}, w) - b (v_{2}, w)

and, symmetrically, all elements enforcing linearity in the second argument. The tensor product is then defined as the quotient

V \otimes W = F (V \times W) / N

with $v \otimes w$ defined as the image of $(v, w)$ in this quotient. Quotienting by $N$ forces exactly the bilinearity relations to hold, and no others, which is what gives the tensor product its universal factorization property.

Basis and Dimension in Finite Dimensions

If $V$ has basis ${e_{1}, \dots, e_{n}}$ and $W$ has basis ${f_{1}, \dots, f_{m}}$ , then the set of simple tensors

{e_{i} \otimes f_{j} ∣ 1 \leq i \leq n, 1 \leq j \leq m}

forms a basis of $V \otimes W$ , so that

\dim (V \otimes W) = \dim (V) \cdot \dim (W) = n m

Every element of $V \otimes W$ can be written as a finite sum of simple tensors $v \otimes w$ , but not every element is itself a simple tensor of this form; elements requiring more than one term in every such expansion are called entangled or non-simple tensors.

Basic Algebraic Properties

The tensor product satisfies bilinearity by construction,

(a v) \otimes w = a (v \otimes w) = v \otimes (a w)

and distributivity over addition in each argument. It is also associative up to canonical isomorphism,

(U \otimes V) \otimes W ≅ U \otimes (V \otimes W)

and commutative up to canonical isomorphism, $V \otimes W ≅ W \otimes V$ , though the isomorphism swaps the order of factors rather than fixing them pointwise. These properties allow the tensor product to be extended unambiguously to any finite number of vector spaces, $V_{1} \otimes \dots \otimes V_{k}$ , without needing to specify an order of pairwise combination.

Relationship to Multilinear Maps

The defining universal property generalizes directly from bilinear to multilinear maps. For vector spaces $V_{1}, \dots, V_{k}$ , every multilinear map

T : V_{1} \times \dots \times V_{k} \to U

factors uniquely through a linear map on $V_{1} \otimes \dots \otimes V_{k}$ . This is precisely the sense in which the tensor product converts multilinear problems into linear ones: rather than reasoning about a function of several vector arguments that is linear in each separately, one reasons about an ordinary linear map defined on a single, larger vector space. The space of multilinear maps from $V_{1} \times \dots \times V_{k}$ to $F$ is accordingly identified with the dual space ${(V_{1} \otimes \dots \otimes V_{k})}^{*}$ , tying the tensor product directly back to the definition of multilinear forms.

Role in Building Tensors

Repeated tensor products of a vector space $V$ and its dual $V^{*}$ generate the full tensor algebra associated to $V$ . A tensor of type $(r, s)$ is, by definition, an element of the tensor product space

\underset{r}{⏨} V \otimes \dots \otimes V \otimes \underset{s}{⏨} V^{*} \otimes \dots \otimes V^{*}

combining $r$ copies of $V$ and $s$ copies of its dual. Because the tensor product operation is exactly what allows several vector spaces to be assembled into one, it is the construction that gives tensors their defining algebraic structure, sitting logically upstream of the notions of multilinear form, symmetric and alternating tensor, and index-based component representation that populate the rest of tensor algebra.