3.1.2 Tensor Dual Space Element Structure

Tensor Dual Space Element Structure is the description of the individual elements that make up a dual tensor space such as V* ⊗ W* or (V*)^{⊗k}, distinguishing simple elements, built directly as ω ⊗ η from single covectors, from general elements, expressed only as finite sums of simple elements, and characterizing each element's rank, its coordinate expression relative to a dual basis, and its interpretation as a multilinear functional acting jointly on several vector arguments. Understanding element structure is what turns the dual tensor space from an abstract vector space into a concrete space of computable, multi-input functionals.

Simple Elements

Definition

A simple element, or decomposable element, of V* ⊗ W* is one of the form ω ⊗ η for a single ω ∈ V* and a single η ∈ W*. As a functional on V × W, a simple element acts by:

which factors as a product of a functional depending only on v and a functional depending only on w. This product structure is the defining feature that distinguishes a simple element from a general one: a simple element's value at (v, w) separates into independent contributions from each argument.

Not Every Element Is Simple

Once dim(V) ≥ 2 and dim(W) ≥ 2, most elements of V* ⊗ W* are not simple; a generic element requires a sum of several simple elements to express. This mirrors the corresponding fact for V ⊗ W itself, and it means a general bilinear form does not, in general, factor as a product of two separate linear functionals.

Rank of a Dual Tensor Space Element

Definition of Rank

The rank of an element β ∈ V* ⊗ W* is the smallest number r of simple elements needed to write β = ω_1 ⊗ η_1 + ⋯ + ω_r ⊗ η_r. A simple element has rank 1 (or rank 0 only for the zero element), and rank increases with the genuine algebraic complexity of the bilinear form β represents.

Rank as Matrix Rank

Relative to bases {e_i} of V and {f_j} of W, an element β = Σ b_{ij} e^i ⊗ f^j corresponds to the matrix (b_{ij}), and the tensor rank of β, in the sense of the minimal decomposition above, equals the ordinary linear-algebra rank of this matrix. This gives a direct computational method for finding the rank of any dual tensor space element built from two factors: form its coordinate matrix and compute the rank by row reduction or any standard method.

Coordinate Expression Relative to a Dual Basis

General Coordinate Form

Every element β ∈ V* ⊗ W* can be written uniquely as β = Σ_{i,j} b_{ij} e^i ⊗ f^j, using the dual bases {e^i} of V* and {f^j} of W* induced from bases {e_i} of V and {f_j} of W. The scalars b_{ij} are recovered by evaluating β on the basis vectors: b_{ij} = β(e_i, f_j), since e^i(e_k) = δ^i_k and f^j(f_l) = δ^j_l isolate exactly the (i, j) term of the sum.

Symmetric and Alternating Elements

When V = W, an element β ∈ V* ⊗ V* is called symmetric if b_{ij} = b_{ji} for all i, j, equivalently if β(v, u) = β(u, v) for all v, u ∈ V, and alternating if b_{ij} = -b_{ji}, equivalently β(v, u) = -β(u, v). These correspond exactly to the elements lying in Sym^2(V*) and Λ^2(V*) respectively, and every element of V* ⊗ V* decomposes uniquely as a sum of a symmetric part and an alternating part.

Elements as Multilinear Functionals of Several Arguments

Higher-Rank Elements

An element of (V*)^{⊗k} acts on k vector arguments jointly, and a simple element ω_1 ⊗ ⋯ ⊗ ω_k again factors as a product ω_1(v_1) ⋯ ω_k(v_k), while a general element is a finite sum of such products. The rank of a higher-order element is defined analogously as the minimal number of simple summands, though unlike the two-factor case, there is no single matrix whose rank computes this quantity directly; higher-order rank is generally harder to determine than the rank of an ordinary matrix.

Evaluation as Repeated Contraction

Evaluating an element T ∈ (V*)^{⊗k} on vector arguments v_1, ..., v_k amounts to contracting each factor of T against the corresponding argument in turn, T(v_1, ..., v_k) = Σ (coordinates of T) · v_1^{i_1} ⋯ v_k^{i_k}, using the coordinates of T relative to the dual basis and the coordinates of each v_l relative to the original basis. This evaluation process is the concrete computational meaning of treating a dual tensor space element as a multilinear functional rather than merely as an abstract tensor.