1.2.30 Tensor Product Space Definition

The tensor product space merges vector spaces, enabling multilinear operations and forming the basis of tensor algebra.

Tensor Product Space Definition is the characterization of the vector space that results from applying the tensor product construction to a collection of vector spaces, considered as an algebraic object in its own right rather than as the operation that produces it. Given vector spaces $V_{1}, \dots, V_{k}$ , their tensor product space is the vector space $V_{1} \otimes \dots \otimes V_{k}$ , whose elements, structure, and dimension are determined entirely by the factor spaces from which it is built.

Formal Description

Given vector spaces $V_{1}, \dots, V_{k}$ over a field $F$ , the tensor product space is the vector space

W = V_{1} \otimes V_{2} \otimes \dots \otimes V_{k}

equipped with the canonical multilinear map $V_{1} \times \dots \times V_{k} \to W$ satisfying the universal property that every multilinear map out of the product $V_{1} \times \dots \times V_{k}$ factors uniquely through a linear map on $W$ . As a vector space in its own right, $W$ supports the ordinary operations of vector addition and scalar multiplication, and questions about its dimension, basis, subspaces, and linear maps can all be posed and answered exactly as for any other vector space, once its relationship to the factor spaces is understood.

Elements of the Tensor Product Space

Every element of the tensor product space is a finite linear combination of elementary tensors $v_{1} \otimes \dots \otimes v_{k}$ , but the space itself is not merely the set of such simple products: it is the vector space they generate, closed under addition and scalar multiplication, and containing sums of elementary tensors that are not themselves elementary. This distinction between the generating elementary tensors and the full ambient space is essential, since most structural questions — dimension, choice of basis, existence of subspaces — concern the space $W$ as a whole rather than any single elementary tensor within it.

Dimension and Basis

Finite-Dimensional Case

When each factor space $V_{i}$ is finite-dimensional with basis ${e_{1}^{i}, \dots, e_{n_{i}}^{i}}$ , the tensor product space $W$ has as a basis the set of all elementary tensors formed by choosing one basis vector from each factor,

{e_{i_{1}}^{1} \otimes \dots \otimes e_{i_{k}}^{k}}

and its dimension is the product of the dimensions of the factor spaces,

\dim (W) = \prod_{i = 1}^{k} \dim (V_{i})

This multiplicative growth in dimension, in contrast to the additive growth of a direct sum, is the defining numerical signature of the tensor product space: two three-dimensional spaces combine to a nine-dimensional tensor product space, whereas their direct sum would only be six-dimensional.

Basis Independence

Although a basis of $W$ is most conveniently exhibited using chosen bases of the factor spaces, the space $W$ itself, along with its dimension and its universal property, does not depend on any particular choice of basis; different choices of bases for the factor spaces yield different, but equally valid, bases of the same underlying tensor product space.

Concrete Realizations

Spaces of Multilinear Maps

The dual of a tensor product space is naturally identified with a space of multilinear maps,

{(V_{1} \otimes \dots \otimes V_{k})}^{*} ≅ Mult (V_{1}, \dots, V_{k}; F)

giving a concrete realization of the tensor product space in terms of a familiar family of functions. This identification is what connects the abstract tensor product space back to the theory of multilinear forms and to component-based tensor calculus.

Spaces of Linear Maps