5.6.5 Tensor Pure Element Generation Role

Tensor Pure Element Generation Role defines how pure elements are created within tensor algebras, forming foundational components for tensor structures and operations.

Tensor Pure Element Generation Role is the function that pure (simple, decomposable) tensors serve as the generating set from which every element of a tensor product space is built. Although a pure tensor is a narrow, highly constrained kind of element, the full tensor product space contains nothing beyond finite linear combinations of such elements, so understanding the generation role means understanding how arbitrary tensors reduce to sums of pure ones.

Formal Statement of the Role

Let V1, V2, …, Vn be vector spaces over a field F and let T denote their tensor product V1 ⊗ V2 ⊗ ⋯ ⊗ Vn. The set of pure tensors

P = {v_{1} \otimes \dots \otimes v_{n} ∣ v_{i} \in V_{i}}

generates T as a vector space, meaning that the F-linear span of P equals all of T. This generating role is not incidental: it is built directly into the construction of the tensor product, which is typically defined as the free vector space on the set of formal symbols v1 ⊗ ⋯ ⊗ vn, modulo the bilinearity (multilinearity) relations that make the tensor product operation well defined.

Why Pure Elements Suffice

A general element of T can involve arbitrarily many terms, yet every one of those terms is itself pure by construction, since the summands arise from the same tensoring operation applied to different tuples of factors.

Construction from the Free Module

In the standard quotient construction of the tensor product, one begins with the free vector space on all tuples (v1, …, vn) and imposes relations that enforce linearity in each coordinate separately. Because the generators of the free vector space are exactly the tuples that become pure tensors after passing to the quotient, the image of every generator is pure, and since the generators span the free vector space, their images span the quotient — which is T itself.

Consequence for Proofs and Definitions

Because pure tensors generate T, many properties of linear maps or bilinear forms defined on T can be verified or defined by specifying behavior only on pure tensors and then extending by linearity. This is the practical payoff of the generation role: it reduces an infinite-dimensional or high-dimensional verification problem to a check on a comparatively simple family of elements.

Generation Versus Uniqueness

The generation role establishes that pure tensors span T, but spanning does not imply that every element has a unique or minimal expression as a sum of pure tensors.

Non-Minimal Sums

Any element of T can be written as a sum of pure tensors in infinitely many different ways, since additional pure terms that cancel each other can always be added without changing the total. The generation role guarantees existence of at least one such sum, not uniqueness of a particular one.

Minimal Sums and Rank

Among all the sums of pure tensors equal to a given element t, the ones using the fewest terms define the tensor rank of t. The generation role guarantees that this minimum is a finite, well-defined number for any t in T, since at least one finite sum of pure tensors always exists to represent t.

Generation Role in the Universal Property

The generation role of pure tensors is the reason the tensor product's universal property is stated the way it is, and the reason that property fully characterizes linear maps out of T.

Determining a Linear Map by Its Values on Pure Tensors

A linear map defined out of T is completely determined once its values on all pure tensors are specified, precisely because those pure tensors generate T. Concretely, if two linear maps f and g from T to a space W agree on every pure tensor, then f and g agree everywhere on T, since any element is a linear combination of pure tensors and both maps are linear.

f (v_{1} \otimes \dots \otimes v_{n}) = g (v_{1} \otimes \dots \otimes v_{n}) \Rightarrow f = g

Bridging Multilinear and Linear Worlds

The universal property associates to every multilinear map φ out of the Cartesian product V1 × ⋯ × Vn a unique linear map φ̂ out of T that agrees with φ on the corresponding pure tensors. It is exactly because pure tensors generate T that this correspondence is a genuine bijection between multilinear maps on the product and linear maps on the tensor product, rather than merely an assignment on part of the space.

Generation Role Across Bases

When each Vi is finite-dimensional, choosing a basis for each factor space produces a distinguished, finite subfamily of pure tensors that generates T even more economically than the full set of all pure tensors.

Basis Tensors as a Minimal Generating Family

If e^(i) ranges over a basis of Vi for each i, the tensors e^(1) ⊗ e^(2) ⊗ ⋯ ⊗ e^(n), formed by choosing one basis vector from each factor, are themselves pure and together form a basis of T, not merely a generating set. This refines the generation role: while all pure tensors generate T, a much smaller, linearly independent selection of pure tensors already suffices, with dimension equal to the product of the dimensions of the factor spaces.

Redundancy Among General Pure Tensors

Because a basis of basis tensors already spans T, the larger family of all pure tensors (with arbitrary, non-basis factors) is highly redundant as a generating set — every pure tensor is itself a linear combination of the finitely many basis tensors, obtained by expanding each factor in the chosen basis and distributing the tensor product.

Illustrative Diagram

The dots mark pure tensors scattered across T; the generation role asserts that every point of the surrounding region, representing an arbitrary element of T, is reachable as a finite linear combination of these marked pure elements.