5.7.4 Tensor Simple Element Span Role

Tensor Simple Element Span Role is the fact, and its consequences, that simple (elementary, rank-one) tensors linearly span the entire tensor product space, so that although no individual simple tensor is a general element, the collection of all simple tensors taken together generates every element through finite linear combination.

Formal Statement

Let V1, V2, …, Vn be vector spaces over a field F and let T = V1 ⊗ V2 ⊗ ⋯ ⊗ Vn. Denote by S the set of all simple tensors,

The span role states that the F-linear span of S equals T:

Equivalently, every element of T can be written as a finite sum of simple tensors with scalar coefficients absorbed into the factors, so that no element of T lies outside every possible linear combination of elements of S.

Origin of the Span Role in the Construction of the Tensor Product

The span role is not a separate theorem requiring independent proof; it is embedded directly in how the tensor product space is constructed.

The Quotient Construction

The standard construction builds T as the quotient of the free vector space on all formal symbols (v1, …, vn) by the subspace of relations enforcing multilinearity. Because the generators of the free vector space map, by definition, to simple tensors under this quotient, and because the generators span the free vector space, their images — the simple tensors — automatically span the quotient space T.

Immediate Consequence

Because span(S) = T falls directly out of the construction, the span role holds unconditionally for every tensor product of vector spaces, regardless of the dimensions of the factor spaces (including the infinite-dimensional case), and regardless of the field F.

Span Role Versus Basis Role

Spanning is a weaker requirement than forming a basis, and the distinction matters for understanding exactly what the simple elements provide.

Simple Tensors Are Not Linearly Independent

The full set S of all simple tensors is far larger, and far more redundant, than any basis of T: many different simple tensors are linear combinations of others (indeed, of the finite set of basis tensors described below), so S spans T without being a linearly independent, let alone minimal, generating set.

A Minimal Spanning Subfamily

When each Vi is finite-dimensional with a chosen basis, the finitely many tensors formed by picking one basis vector from each factor are themselves simple, and this finite subfamily is not just a spanning set but an actual basis of T, of size equal to the product of the factor dimensions. This shows that although the full set S is needed for the unconditional, basis-independent statement of the span role, a much smaller, carefully chosen subset of S already suffices once bases are available.

Practical Uses of the Span Role

The span role converts many general statements about tensors into statements that need only be checked on simple tensors, which is its chief practical value.

Defining Linear Maps by Their Action on Simple Tensors

Because simple tensors span T, a linear map out of T is completely determined by specifying its values on all simple tensors (and, in fact, only needs to be specified consistently on a basis drawn from simple tensors). This is the mechanism underlying the universal property of the tensor product, which builds a linear map φ̂ on T out of a multilinear map φ by declaring φ̂'s values on simple tensors and relying on the span role to extend that declaration, by linearity, to the whole of T.

Proving Identities by Reduction to Simple Tensors

An identity or inequality claimed to hold for every element of T can often be proved by verifying it first on simple tensors and then invoking linearity (and, where relevant, continuity or other structure) to extend the result to arbitrary elements, since the span role guarantees no element escapes this reduction.

Approximation and Decomposition

Numerical tensor decomposition methods rely on the span role at a foundational level: expressing a target tensor as an approximate or exact sum of few simple tensors is only a sensible objective because the span role guarantees that an exact sum of simple tensors — possibly using very many terms — always exists; the practical question decomposition methods address is how few terms are actually needed.

Illustrative Diagram

The dashed lines suggest linear combinations reaching between the marked simple tensors, illustrating that the span role guarantees any point in the surrounding space is expressible this way, even though the simple tensors themselves occupy only a thin, curved subset.