2.8.2 Tensor Span Closure Property

The Tensor Span Closure Property ensures spanning sets in tensor algebras remain closed under tensor operations, forming a key structure in multilinear algebra.

Tensor Span Closure Property is the statement that the set of all linear combinations of a given collection of vectors from a vector space used in tensor construction forms a subset that is stable under the two operations that define a vector space: vector addition and scalar multiplication. If a collection of vectors is drawn from a vector space over a field, the span of that collection is the smallest subspace containing every vector in the collection, and the closure property guarantees that combining span elements by addition or scaling them by field elements never produces a result outside the span.

Formal Statement

Span as a Set of Linear Combinations

Given a vector space over a field and a finite or arbitrary subset of vectors drawn from that space, the span consists of every vector expressible as a finite linear combination of vectors from the subset, with coefficients taken from the field.

span (S) = {\sum_{i = 1}^{n} c_{i} v_{i} ∣ c_{i} \in F, v_{i} \in S}

Closure Under Addition

For any two vectors that belong to the span of a set, their sum also belongs to the span. This follows directly because the sum of two linear combinations of vectors from the set is itself a linear combination of vectors from the same set, obtained by combining the coefficient lists.

u, w \in span (S) \Rightarrow u + w \in span (S)

Closure Under Scalar Multiplication

For any vector in the span of a set and any scalar drawn from the underlying field, the scaled vector also belongs to the span. Scaling a linear combination by a field element distributes across every term, producing another valid linear combination of the same generating vectors.

u \in span (S), λ \in F \Rightarrow λ u \in span (S)

Consequence: Span Is a Subspace

Subspace Criterion

A subset of a vector space qualifies as a subspace precisely when it contains the zero vector, is closed under addition, and is closed under scalar multiplication. The span of any set automatically satisfies all three conditions, since the zero vector arises from the trivial linear combination with every coefficient equal to zero, and the two closure properties are established directly from the structure of linear combinations.

Independence from the Generating Set's Size

The closure property holds regardless of whether the generating set is finite, countably infinite, or uncountable, because every individual linear combination used in a closure argument still involves only finitely many terms with nonzero coefficients. Infinite generating sets do not require infinite sums to remain inside the span.

Role in Tensor Basis Construction

Support for Basis Span Structure

Within the structure of a tensor basis span, the closure property is what allows any spanning set of basis vectors to generate a well-defined subspace that tensor construction can rely on. Without closure, sums and scaled combinations of basis vectors could fall outside the intended structure, breaking the coherence of coordinate representations built from that basis.

Compatibility with Multilinear Extension

When vector spaces satisfying the closure property are combined to build tensor products, the resulting tensor spaces inherit an analogous closure behavior with respect to the multilinear combinations used to define tensors, since each factor space already guarantees that linear combinations remain confined to its own span.

Summary of Key Properties

Stability Under the Two Vector Space Operations

The span closure property certifies that no operation native to the vector space, namely addition of two span elements or multiplication of a span element by a scalar, can produce a vector lying outside the span, making the span a self-contained algebraic structure.

Foundational Status

Because this closure property is what elevates a mere set of linear combinations into a genuine subspace, it functions as a foundational fact that later constructions in tensor algebra, including basis coverage, coordinate representation, and dimension counting, depend upon implicitly.