3.13.2 Tensor Canonical Embedding Evaluation

Tensor Canonical Embedding Evaluation is the underlying pairing operation between a vector space V and its dual V* that supplies the raw scalar output used to build the canonical embedding of V into V**. Where the embedding itself is a map from vectors to double-dual functionals, the evaluation is the more elementary bilinear operation, taking a covector and a vector together and returning a scalar, out of which the entire embedding is assembled. Understanding evaluation as an operation in its own right clarifies why the embedding is canonical: it is built from nothing more than the pairing already present between V and V*.

The Evaluation Pairing

Basic Definition

The evaluation pairing is the map that takes a covector φ in V* and a vector v in V and returns the scalar φ(v), obtained simply by applying the linear functional φ to the vector v:

This map is often written using angle bracket notation, ⟨φ, v⟩, to emphasize its symmetric, pairing-like character even though φ and v come from different spaces.

Bilinearity of Evaluation

Evaluation is linear in each of its two arguments separately, holding the other fixed. Linearity in φ follows because V* inherits its vector space structure pointwise: (aφ + bψ)(v) = aφ(v) + bψ(v). Linearity in v follows because each φ is itself a linear functional: φ(av + bw) = aφ(v) + bφ(w). Together these two facts make evaluation a bilinear map on V* × V.

From Evaluation to the Embedding

Fixing the Vector Argument

The canonical embedding map ev: V → V** arises from the evaluation pairing by fixing the vector argument v and allowing the covector argument to vary. For fixed v, the partial function φ ↦ Eval(φ, v) is linear in φ, by the linearity of evaluation in its first argument, and this partial function is exactly the functional ev(v) in V**:

This shows that the embedding is not an independent construction but a direct repackaging of the evaluation pairing, obtained by treating v as a parameter and φ as the variable.

Why This Construction Requires No Extra Choices

Because evaluation is already defined the moment V* is defined as the space of linear functionals on V, fixing v and reading off the resulting function of φ requires no additional data, such as a basis or an inner product. This is the precise sense in which the embedding is canonical: it is a direct consequence of the definitions of V* and evaluation, rather than an extra structure imposed afterward.

Nondegeneracy of Evaluation

Statement of Nondegeneracy

The evaluation pairing is nondegenerate in its vector argument on a finite-dimensional space V: if v is a nonzero vector, there exists some φ in V* with Eval(φ, v) ≠ 0. This property is what guarantees the embedding ev is injective, since a nonzero v cannot be sent to the zero functional in V** if some covector detects it as nonzero.

Constructing a Detecting Covector

A covector that detects a given nonzero vector v can always be constructed explicitly: extend v to a basis v = e_1, e_2, ..., e_n of V, form the dual basis e^1, ..., e^n, and take φ = e^1. By the defining property of the dual basis, e^1(e_1) = 1, so Eval(e^1, v) = 1 ≠ 0. This explicit construction confirms nondegeneracy is not merely an abstract existence claim but something that can be exhibited concretely in any finite-dimensional space.

Nondegeneracy in the Other Argument

The evaluation pairing is also nondegenerate in its covector argument: if φ is a nonzero covector, there exists some v in V with φ(v) ≠ 0, simply because φ being nonzero as a functional means it does not vanish on every vector. This symmetric nondegeneracy underlies the fact that V and V* are placed on an equal footing by the evaluation pairing, even though the embedding built from it goes specifically from V into V**.

Evaluation and Component Computation

Evaluation in Coordinates

Given a basis e_1, ..., e_n of V with dual basis e^1, ..., e^n, a vector v = v^i e_i and a covector φ = φ_j e^j, written with the Einstein summation convention, evaluation reduces to a single sum over matching indices:

This coordinate formula is the practical, computational face of the evaluation pairing, reducing an abstract pairing between two spaces to an ordinary dot-product-like sum of matched components.

Independence of the Formula from the Chosen Basis

Although the formula for evaluation in coordinates references a specific basis and its dual, the scalar value φ(v) produced does not depend on which basis was used to compute it. Any other basis and its corresponding dual basis will yield the same numerical result, since evaluation is defined abstractly before any coordinates are introduced, and the coordinate formula is merely one way of computing an already basis-independent quantity.

Diagrammatic Summary

The diagram shows the two inputs, a covector φ and a vector v, feeding into the evaluation pairing, which outputs a scalar in the field F. Fixing v and treating the output as a function of φ alone is exactly the operation that produces the canonical embedding of v into V**.