3.1.5 Tensor Dual Space Vector Space Relation

Tensor Dual Space Vector Space Relation is the precise account of how a dual tensor space such as V* ⊗ W* relates to the vector space category as a whole, establishing that it is itself an ordinary finite-dimensional vector space, subject to every general theorem about vector spaces, while also carrying the extra multilinear-functional meaning inherited from its construction out of V* and W*. This relation is what licenses applying standard linear-algebra tools, bases, linear independence, subspaces, quotient spaces, direct sum decompositions, directly to dual tensor spaces, with the understanding that these tools coexist with, rather than replace, the tensor-specific structure already described elsewhere.

The Dual Tensor Space as an Ordinary Vector Space

Satisfying the Vector Space Axioms

V* ⊗ W*, built as the tensor product of two vector spaces, is itself a vector space over the same field F: addition of elements is commutative and associative, the zero element 0 is an additive identity, every element has an additive inverse, and scalar multiplication distributes over addition, satisfies 1 · β = β, and is compatible with field multiplication, (cd)β = c(dβ). None of these properties require any reference to the multilinear-functional interpretation of V* ⊗ W*; they follow directly from the vector-space axioms already established when the tensor product was constructed.

Applicability of General Vector-Space Theorems

Because V* ⊗ W* is an ordinary vector space, general facts, that any two bases have equal cardinality, that a linearly independent set extends to a basis, that a linear map is determined by its values on a basis, that a subspace has a well-defined dimension, apply to V* ⊗ W* exactly as they apply to any other vector space, with no modification needed to accommodate its tensor-product origin.

Where the Extra Structure Comes From

Two Layers of Meaning

V* ⊗ W* carries two layers of structure at once: as a vector space, it has addition, scalar multiplication, dimension, and subspaces like any other; as a tensor space built from V* and W* specifically, it additionally carries the tensor product operation relating it to V*, to W*, and to other tensor spaces, the identification with bilinear functionals on V × W, and the notion of simple elements and tensor rank. The vector-space relation is the first layer, present in every vector space; the specifically tensorial content is the second layer, present only because of how V* ⊗ W* was constructed.

Simple Elements Are Not a Vector-Space Notion

The property of being a simple element, ω ⊗ η for single covectors ω and η, is meaningful only relative to the tensor-product construction of V* ⊗ W* from its factors V* and W*; an abstract vector space with no distinguished tensor decomposition has no analogous notion of "simple" elements. This illustrates precisely how the tensor-specific structure goes beyond what the ordinary vector-space relation alone provides.

Subspaces of Dual Tensor Spaces

Symmetric and Alternating Subspaces as Ordinary Subspaces

The subspaces Sym^2(V*) and Λ^2(V*) of V* ⊗ V*, defined by the symmetry conditions on the corresponding bilinear forms, are ordinary vector subspaces in the standard sense, closed under addition and scalar multiplication, and satisfying V* ⊗ V* = Sym^2(V*) ⊕ Λ^2(V*) as an ordinary direct sum decomposition of vector spaces. This decomposition is simultaneously a fact about the vector-space structure of V* ⊗ V*, since direct sum decomposition is a general vector-space notion, and a fact reflecting the specifically tensorial symmetry properties of its elements.

Rank Strata Are Not Subspaces

By contrast, the set of elements of V* ⊗ W* having tensor rank exactly r, or at most r, is generally not a vector subspace, since, as established in the discussion of addition structure, the sum of two rank-r elements can have rank different from r. This is a clear instance where a tensor-specific notion, rank, fails to interact with the vector-space relation in the same well-behaved way that a genuinely linear notion, such as membership in Sym^2 or Λ^2, does.

Isomorphism as Vector Spaces Versus Isomorphism Respecting Tensor Structure

Two Different Standards of Equivalence

Two dual tensor spaces of equal dimension are isomorphic merely as vector spaces, by the general theorem that dimension is a complete invariant, but this says nothing about whether a natural isomorphism exists that also respects the tensor product, the identification with multilinear functionals, and the notion of simple elements. The vector-space relation supplies the weaker, dimension-based equivalence; the specifically tensorial structure supplies the stronger, structure-preserving equivalence already discussed for tensor spaces built directly from V.

Practical Consequence

When working with a dual tensor space, it is necessary to keep track of which claims rely only on the vector-space relation, true of any vector space of the same dimension, and which claims rely on the specific tensor-product construction, true only because V* ⊗ W* was built in this particular way from V* and W*; conflating the two is a common source of error when reasoning about tensor spaces.