3.15.5 Tensor Dual Map Identity Preservation

Tensor Dual Map Identity Preservation is the property of the dual map construction whereby dualizing the identity map on a vector space returns the identity map on its dual space, so that (id_V)* = id_{V*} for every finite-dimensional vector space V. This fact anchors the dual map construction as a genuine functor rather than an arbitrary assignment: together with the order-reversal law for composition, identity preservation supplies the second of the two conditions required for the assignment V ↦ V*, f ↦ f* to satisfy the axioms of a contravariant functor, and it underlies why dualizing leaves the "do-nothing" transformation unchanged.

Statement and Proof

The Identity

For any finite-dimensional vector space V with identity map id_V: V → V, the dual construction yields

meaning the dual of the identity map on V is the identity map on V*, not merely a map isomorphic to it, but the literal identity transformation of the dual space.

Direct Verification

The proof follows immediately from the defining property of dual maps through precomposition. For any covector φ ∈ V* and any vector v ∈ V,

Since (id_V)*(φ) applied to any v returns exactly φ(v), the covector (id_V)*(φ) is φ itself. This holds for every φ ∈ V*, so (id_V)* acts as the identity on all of V*, establishing the claim without appeal to bases or matrices.

Matrix-Level Confirmation

Transpose of the Identity Matrix

Relative to any basis of V and the corresponding dual basis of V*, the identity map id_V is represented by the identity matrix I. Since the dual map corresponds to matrix transposition, (id_V)* is represented by I^T, and because the identity matrix is symmetric, I^T = I. The matrix computation reproduces the abstract result: the dual of the identity is represented by the identity matrix in the dual basis.

Basis Independence of the Result

Identity preservation holds regardless of which basis is chosen for V, because the identity map has the same matrix representation, the identity matrix, in every basis, and the transpose of the identity matrix is the identity matrix in every basis as well. The result is therefore a statement about the abstract linear map, not an artifact of a particular coordinate choice.

Role in the Functor Axioms

The Two Functor Conditions

A contravariant functor from vector spaces to vector spaces must satisfy two conditions: it must reverse composition, (g ∘ f)* = f* ∘ g*, and it must preserve identities, (id_V)* = id_{V*}. Identity preservation is precisely the second of these two conditions, and without it the dual construction would fail to qualify as a functor even if the composition law held.

Interaction with Composition Reversal

Identity preservation and composition reversal work together to guarantee that dual maps behave consistently across chains of transformations. For instance, applying the composition law to f ∘ id_V = f gives (id_V)* ∘ f* = f*, which is only consistent because (id_V)* acts as the identity on V*, leaving f* unchanged when composed with it.

Consequence for Invertible Maps

Identity preservation is the key ingredient in showing that duals of inverses are inverses of duals. From f ∘ f^{-1} = id_W and f^{-1} ∘ f = id_V, dualizing both equations and applying identity preservation on each side yields (f^{-1})* ∘ f* = id_{W*} and f* ∘ (f^{-1})* = id_{V*}, which together show that f* is invertible with inverse (f^{-1})*.

Identity Preservation on Tensors

Trivial Action on the Tensor Algebra

When the identity map id_V is used to transform a type (p, q) tensor by acting with id_V on each contravariant slot and (id_V)* = id_{V*} on each covariant slot, every slot is left unchanged, so the tensor itself is left unchanged. Identity preservation guarantees that the "do-nothing" transformation on V induces the "do-nothing" transformation on every tensor space built from V and V*, with no exceptions arising from the covariant slots.

Anchoring the Group Action

In contexts where invertible endomorphisms of V act on tensors of type (p, q) via the combined action on contravariant and covariant slots, identity preservation ensures that this action satisfies the basic requirement of a group action: the identity element of the group of invertible maps acts as the identity transformation on the space of tensors, which is a prerequisite for the action to be well-defined at all.

Diagrammatic Summary

The loop on the left represents the identity map id_V looping from V back to V, and the loop on the right represents its dual, which loops from V* back to V* as the identity id_{V*}, illustrating that dualizing the identity leaves the loop structure unchanged.