3.8.2 Tensor Natural Pairing Dual Evaluation

Tensor Natural Pairing Dual Evaluation is the reversed reading of the natural pairing in which a vector, rather than a covector, is treated as the object performing the evaluation, acting on a covector through the canonical identification of V with the double dual space V**. Where the ordinary evaluation f(v) treats f as a function applied to v, dual evaluation treats v, reinterpreted as an element of V**, as a function applied to f, and the natural pairing guarantees that both readings always agree numerically.

Setting Up Dual Evaluation

The Double Dual Space

The double dual V** is the dual space of V*, consisting of all linear functionals on V*. Since V* is itself a vector space, V** is defined exactly the same way any dual space is defined, simply applied one level higher: an element of V** is a linear map V* -> F.

The Canonical Embedding Revisited

The canonical map ι : V -> V** sends each vector v to the functional ι(v) defined by

for every f in V*. Dual evaluation is the act of applying ι(v) to f, written v(f) by identifying v with ι(v), and by definition it produces the same scalar as the ordinary evaluation f(v).

Why Dual Evaluation Is Well-Defined

Linearity of the Dual Evaluation Map

For ι(v) to genuinely belong to V**, it must be a linear functional on V*, meaning it must satisfy additivity and homogeneity with respect to its covector argument. Both properties follow directly from the vector space structure of V*:

since covector addition and scalar multiplication are themselves defined pointwise using the underlying evaluation on V.

Injectivity and Surjectivity in Finite Dimensions

When V is finite-dimensional, ι is both injective, meaning distinct vectors give distinct functionals on V*, and surjective, meaning every functional on V* arises from some vector in V. This makes ι a genuine isomorphism, and it is precisely this isomorphism that licenses treating v(f) and f(v) as interchangeable notations for the same scalar.

Dual Evaluation in Components

Symmetric Coordinate Formula

Relative to a basis e_1, ..., e_n of V and its dual basis e^1, ..., e^n of V*, dual evaluation gives the identical summation formula as ordinary evaluation:

There is no separate coordinate formula to learn for dual evaluation; it reuses exactly the same component computation, simply interpreted with the roles of subject and object swapped conceptually.

The Double Dual Basis

Under the identification V ≅ V**, the basis e_1, ..., e_n of V corresponds to a basis of V** that is precisely the dual basis of e^1, ..., e^n, closing the loop: taking the dual of the dual basis returns the original basis, up to this canonical identification.

Significance of Dual Evaluation

Symmetry Between V and V*

Dual evaluation is what makes the relationship between V and V* fully symmetric in finite dimensions: each space can be regarded as the dual of the other, and each object can be regarded as acting on, or being acted upon by, the other. Neither V nor V* is intrinsically the more fundamental space; the natural pairing treats them evenhandedly.

Infinite-Dimensional Caveat

The symmetry underlying dual evaluation depends on ι being an isomorphism, which holds automatically only in finite dimensions. In infinite-dimensional vector spaces, ι is generally injective but not surjective, so V embeds into V** without necessarily equaling it, and dual evaluation must then be understood as evaluation of the embedded copy of V, not of the full space V**.

Diagrammatic Summary

The diagram shows the vector, reinterpreted through ι as an element of V**, evaluating a covector, producing the same scalar the covector would produce evaluating the original vector.