3.2 Tensor Covector Structure

Tensor Covector Structure is the set of algebraic properties governing covectors specifically as the lower-index building blocks of general tensor spaces, covering how covectors combine via the tensor product to produce purely covariant tensors, how a covector acts as a contraction partner against a vector index of any higher tensor, and how the space of covectors itself, V*, supports the full range of tensor operations, symmetrization, antisymmetrization, and tensor product, in exact parallel with the corresponding operations on V. Covector structure is what makes it possible to build an entire tensor algebra, T(V*) = ⊕_k (V*)^{⊗k}, from covectors alone, mirroring the tensor algebra built from V.

Purely Covariant Tensors From Covectors

The Space (V*)^{⊗k}

The k-fold tensor product (V*)^{⊗k}, consisting of simple elements ω_1 ⊗ ⋯ ⊗ ω_k and their sums, is the space of purely covariant, (0, k)-tensors on V. Every element of this space is, by the multilinear functional identification, a k-linear map V × ⋯ × V → F, taking k vector arguments and combining them into a single scalar.

Basis and Dimension

Given a basis {e_i} of V with dual basis {e^i} of V*, the simple tensors {e^{i_1} ⊗ ⋯ ⊗ e^{i_k}}, indexed over all choices of k indices from 1 to n = dim(V), form a basis of (V*)^{⊗k}, giving dim((V*)^{⊗k}) = n^k, matching the general tensor power dimension formula applied to the carrier space V*.

Covectors as Contraction Partners

Contraction With a Vector

A single covector ω ∈ V* pairs directly with a single vector v ∈ V through the natural pairing ω(v), and this pairing is the basic instance of contraction: whenever a general tensor T has an upper (vector-valued) index, a covector can be contracted against that index to reduce the tensor's rank by one. For a (p, q)-tensor T and a covector ω, contraction against the first upper index produces a (p-1, q)-tensor:

using the coordinates ω_i of ω relative to the dual basis and the components T^i(...) of T in its first upper slot. This is the direct generalization of applying a single covector to a single vector.

Covectors as the Building Blocks of All Lower Indices

Every lower index of a general (p, q)-tensor is, by construction, a factor drawn from V*, so covector structure, in particular the transformation rule for covector components and the natural pairing with vectors, is inherited unchanged by every lower index of every tensor built from V; there is no separate theory needed for lower indices of higher tensors beyond the theory of covectors themselves, applied slot by slot.

Symmetric and Alternating Covector Tensors

Symmetric Covector Tensors

The subspace Sym^k(V*) ⊂ (V*)^{⊗k} consists of covariant tensors invariant under permutation of their k vector arguments, T(v_1, ..., v_k) = T(v_{σ(1)}, ..., v_{σ(k)}) for every permutation σ. These correspond to symmetric multilinear forms, such as the quadratic forms and inner-product-like pairings that appear throughout linear algebra, and by the dual-power isomorphism, Sym^k(V*) ≅ (Sym^k(V))*.

Alternating Covector Tensors: Differential Forms

The subspace Λ^k(V*) ⊂ (V*)^{⊗k} consists of covariant tensors that change sign under transposition of any two arguments. Elements of Λ^k(V*) are exactly the algebraic prototype of differential k-forms, the alternating multilinear objects used to define volume, area, and orientation on a vector space, and dim(Λ^k(V*)) = C(n, k) follows the same binomial dimension count established for exterior powers built from V directly.

The Covector Tensor Algebra

Direct Sum Structure

Collecting all covariant tensor powers into a single graded vector space, T(V*) = F ⊕ V* ⊕ (V*)^{⊗2} ⊕ (V*)^{⊗3} ⊕ ⋯, equipped with the tensor product as multiplication, forms the covariant tensor algebra of V, a graded algebra in which the product of a j-fold covector tensor and a k-fold covector tensor is a (j+k)-fold covector tensor. This algebra is the exact structural mirror of the contravariant tensor algebra T(V) = ⊕_k V^{⊗k}, built entirely from covectors instead of vectors, and its existence confirms that V* supports the full generality of tensor algebra operations already established for V.