3.1 Tensor Dual Space Structure

Tensor Dual Space Structure is the extension of the dual-space construction from a single vector space V to the tensor spaces built from V, describing how the dual of a tensor product, (V ⊗ W)*, relates to the tensor product of duals, V* ⊗ W*, and how this relationship underlies the general fact that every tensor space T^p_q(V) is itself naturally the dual of another tensor space built from V and V*. This structure is what allows covectors, bilinear forms, and general (p, q)-tensors to be treated uniformly as multilinear functionals, rather than as separate constructions each requiring its own foundation.

The Dual of a Tensor Product

Natural Isomorphism (V ⊗ W)* ≅ V* ⊗ W*

For finite-dimensional V and W, there is a canonical isomorphism between the dual of the tensor product V ⊗ W and the tensor product of the duals V* ⊗ W*. A simple tensor ω ⊗ η ∈ V* ⊗ W*, for ω ∈ V* and η ∈ W*, corresponds to the linear functional on V ⊗ W defined on simple tensors by:

and extended bilinearly. Because a basis {e^i ⊗ f^j} of V* ⊗ W* maps to a basis of (V ⊗ W)* dual to the standard basis {e_i ⊗ f_j} of V ⊗ W, this correspondence is a full vector-space isomorphism, not merely an injection.

Bilinear Forms as the Concrete Meaning

Because a linear functional on V ⊗ W is, by the universal property of the tensor product, the same data as a bilinear map V × W → F, the isomorphism (V ⊗ W)* ≅ V* ⊗ W* says exactly that every bilinear form on V × W is representable as a tensor built from covectors on V and covectors on W, with the simple tensors ω ⊗ η corresponding to the elementary, rank-one bilinear forms (v, w) ↦ ω(v)η(w).

Duals of Higher Tensor Powers

The General Pattern

Iterating the construction gives (V^{⊗k})* ≅ (V*)^{⊗k} for any k, so the dual of the k-fold tensor power of V is naturally isomorphic to the k-fold tensor power of the dual of V. Under this isomorphism, a simple tensor ω_1 ⊗ ⋯ ⊗ ω_k ∈ (V*)^{⊗k} acts on a simple tensor v_1 ⊗ ⋯ ⊗ v_k ∈ V^{⊗k} by:

so k-linear forms on V^{×k} correspond exactly to tensors in (V*)^{⊗k}, generalizing the two-factor bilinear form case.

Duals of Symmetric and Alternating Powers

Restricting attention to symmetric tensors, (Sym^k(V))* ≅ Sym^k(V*), so symmetric multilinear forms on V correspond to symmetric tensors built from V*; similarly (Λ^k(V))* ≅ Λ^k(V*), so alternating multilinear forms correspond to alternating tensors built from V*. These restrictions are consistent with the general tensor-power isomorphism because the symmetrization and antisymmetrization operations commute with dualizing.

General (p, q)-Tensors as Multilinear Functionals

The Standard Identification

The tensor space T^p_q(V) = V^{⊗p} ⊗ (V*)^{⊗q} used to describe tensors with p upper and q lower indices is naturally isomorphic to the space of multilinear functionals taking p covector arguments and q vector arguments and producing a scalar:

A (p, q)-tensor T = v_1 ⊗ ⋯ ⊗ v_p ⊗ ω_1 ⊗ ⋯ ⊗ ω_q acts on p covectors η^1, ..., η^p and q vectors u_1, ..., u_q by pairing each v_i with η^i and each ω_j with u_j, and multiplying all the resulting scalars together. This is the sense in which every tensor, regardless of its variance type, is fundamentally a multilinear functional built from V and V* in combination.

Why Lower Indices Pair With Vector Arguments

Because a (1, 0)-tensor is a vector v ∈ V, and a vector applied to a covector argument η gives η(v), the natural pairing already described for V and V*, the upper index of a tensor corresponds to a slot that accepts a covector argument, while the lower index, contributed by a factor of V*, corresponds to a slot that accepts a vector argument; this is the origin of the convention that lower indices consume vectors and upper indices consume covectors when a tensor is regarded as a multilinear functional.

Consistency With Coordinate Transfer and Structure Preservation

Matching Variance Conventions

The identification of T^p_q(V) with multilinear functionals is fully consistent with the coordinate transfer rules for (p, q)-tensors and with the structure-preservation properties of induced isomorphisms: an upper index, built from a factor of V, transforms contravariantly, and a lower index, built from a factor of V*, transforms covariantly, exactly matching the transformation rules already established for vector and covector components individually, now applied factor by factor across the whole tensor.