3.1.1 Tensor Dual Space Carrier Structure

Tensor Dual Space Carrier Structure is the description of how a dual tensor space such as V* ⊗ W* or (V*)^{⊗k} depends functorially on its carrier spaces, the original vector spaces V and W from which the dual factors are built, covering how a linear map between carrier spaces induces a corresponding map between their dual tensor spaces, how the carrier spaces determine the dimension and basis of the dual tensor space, and how the direction of induced maps reverses when passing from a carrier space to its dual. The carrier is the source of all structure in a dual tensor space: nothing about V* ⊗ W* is specified independently of V and W, and every property of the dual tensor space can be traced back to a corresponding property of its carrier spaces.

Carrier Spaces and the Contravariant Functor

Reversal of Map Direction

If φ : V → U is a linear map between carrier spaces, the induced map on duals, the pullback φ* : U* → V*, defined by φ*(η) = η ∘ φ for η ∈ U*, goes in the opposite direction from φ itself. This reversal is intrinsic to dualizing: since a covector on U consumes vectors from U, and φ produces elements of U from elements of V, composing η after φ produces a covector on V, not on U.

Contravariant Functoriality

The pullback satisfies (ψ ∘ φ)* = φ* ∘ ψ* for composable carrier maps φ : V → U and ψ : U → X, with the order of composition reversed relative to ψ ∘ φ, and (id_V)* = id_{V*}. This pattern, direction-reversing on maps while respecting composition in reverse order, is what it means for the dual-space construction to be a contravariant functor of its carrier, in contrast with the covariant, direction-preserving functoriality of tensor powers of V itself.

Carrier Structure of Multi-Factor Dual Tensor Spaces

Pullback on V* ⊗ W*

For carrier spaces V and W with linear maps φ : V → U and ψ : W → X, the induced map on the dual tensor space V* ⊗ W* is built from the pullbacks φ* : U* → V* and ψ* : X* → W*, combined as φ* ⊗ ψ* : U* ⊗ X* → V* ⊗ W*, sending a simple tensor η ⊗ ζ to φ*(η) ⊗ ψ*(ζ) = (η ∘ φ) ⊗ (ζ ∘ ψ). The overall direction reversal, from a pair of carrier maps V → U and W → X to a single dual map U* ⊗ X* → V* ⊗ W*, matches the single-factor case applied independently to each carrier factor.

Multilinear Functional Interpretation

Under the identification of V* ⊗ W* with bilinear forms on V × W, the pullback φ* ⊗ ψ* corresponds exactly to precomposing a bilinear form β : U × X → F with φ and ψ in each argument, (φ, ψ)*β)(v, w) = β(φ(v), ψ(w)), which is the natural way to transport a bilinear form back along carrier maps, confirming that the tensor-level construction and the functional-level construction of carrier structure agree.

Dimension and Basis Dependence on the Carrier

Dimension Determined by the Carrier

Because dim(V*) = dim(V), the dimension of any dual tensor space built from carriers V, W, ... is determined entirely by the dimensions of the carrier spaces themselves, following the same multiplicative and additive counting rules that govern ordinary tensor constructions: dim(V* ⊗ W*) = dim(V) × dim(W), and more generally dim((V*)^{⊗k}) = dim(V)^k.

Basis Induced From the Carrier

A basis of a dual tensor space is never chosen independently; it is always the dual basis induced from a chosen basis of the carrier, {e^i ⊗ f^j} built from {e_i} and {f_j}, and a change of basis in the carrier propagates automatically to a corresponding, uniquely determined change of the dual basis via the covariant transformation rule. The carrier is therefore not just the algebraic source of a dual tensor space's elements, but also the sole source of any coordinate system used to describe them.

Carrier Structure and Isomorphisms

Isomorphic Carriers Give Isomorphic Duals

If φ : V → U is an isomorphism of carrier spaces, the pullback φ* : U* → V* is also an isomorphism, with inverse (φ^{-1})*, since φ* ∘ (φ^{-1})* = (φ^{-1} ∘ φ)* = (id_V)* = id_{V*} and symmetrically for the other composite. This shows that the dual-tensor-space construction preserves isomorphism, in the same structure-preserving sense established for ordinary tensor constructions, even though the direction of the induced map is reversed relative to the direction of the original carrier isomorphism.

Non-Uniqueness of the Carrier for a Given Dual Space

Two different carrier spaces of the same dimension produce abstractly isomorphic dual tensor spaces, but, exactly as for V and V* themselves, this isomorphism is not canonical unless a specific isomorphism of the carriers is fixed; carrier structure is what supplies the extra data, a specific map between carriers, needed to pin down a specific, natural map between the corresponding dual tensor spaces, rather than merely an abstract isomorphism justified by matching dimension.