3.6 Tensor Covector Evaluation Operation

Tensor Covector Evaluation Operation is the pairing rule that produces a scalar by applying a covector, an element of a dual space, to a vector belonging to the original vector space, and it is the single bilinear operation from which every higher tensor contraction involving vectors and covectors is built. Written <f, v> or f(v), the operation takes a linear functional f in V* and a vector v in V and returns an element of the underlying field, and it is linear in both of its arguments simultaneously, which is what makes it the prototypical example of a (1, 1)-type pairing in tensor algebra.

Formal Definition

The Pairing Map

Let V be a vector space over a field F, and let V* denote its dual space, the set of all linear functionals f : V -> F. The tensor covector evaluation operation is the map

defined by

for every f in V* and every v in V. The left-hand notation <f, v> treats the operation as a bilinear pairing between two spaces, while the right-hand notation f(v) treats it as ordinary function application, and both notations describe exactly the same value.

Bilinearity

The evaluation operation is linear in the covector argument,

and linear in the vector argument,

for scalars a, b in F. Because it is linear in each slot separately while holding the other slot fixed, the evaluation operation is precisely a (1, 1) tensor: it consumes one covector and one vector and produces a scalar.

Component Form

Index Notation

Given a basis e_1, ..., e_n of V and its dual basis e^1, ..., e^n of V*, satisfying the defining relation <e^i, e_j> = &delta;^i_j, any covector f = f_i e^i and any vector v = v^j e_j evaluate to

using the Einstein summation convention, where the repeated index i is summed from 1 to n. The covariant components f_i of the covector carry lower indices, the contravariant components v^i of the vector carry upper indices, and the evaluation operation contracts the shared index, leaving no free indices, which is why its output is a scalar rather than another tensor.

The Kronecker Delta as the Basis Pairing

The relation <e^i, e_j> = &delta;^i_j states that each dual basis covector e^i returns 1 when evaluated on the matching basis vector e_j and 0 on every other basis vector:

This relation is what defines the dual basis uniquely once a basis of V is fixed, and it is the mechanism through which the abstract evaluation operation reduces to the ordinary dot-product-like sum f_i v^i in coordinates.

Relationship to Tensor Contraction

Evaluation as the Simplest Contraction

The evaluation operation is the base case of the general tensor contraction operation. Contraction always pairs one upper index of a tensor with one lower index and sums over it using the same mechanism as f_i v^i. When the two tensors being contracted are a single covector and a single vector, the contraction produces exactly the evaluation pairing, with no free indices remaining. Higher contractions, such as those applied to a (1, 1) tensor to compute its trace, or those applied to a (p, q) tensor along one chosen pair of indices, all reduce locally to this same upper-lower index sum.

Composition with Multilinear Maps

Because a general (1, 1) tensor can be identified with a linear map T : V -> V, evaluating a covector against Tv for a vector v, written <f, Tv>, gives a scalar that depends linearly on f, T, and v simultaneously. This triple pairing underlies operations such as computing matrix entries: the (i, j) entry of the matrix representing T in a given basis equals <e^i, T e_j>, directly using the evaluation operation to extract each component.

Independence from Choice of Basis

Coordinate-Free Character

Although the component formula f_i v^i depends on a choice of basis, the scalar value <f, v> it computes does not. If the basis is changed by an invertible matrix A, the components of f transform by A and the components of v transform by A^{-1}, or vice versa depending on convention, and these two transformations cancel exactly in the contracted sum, leaving the evaluated scalar unchanged. This cancellation is the defining property that distinguishes a legitimate covector-vector pairing from an arbitrary bilinear combination of components.

Why the Pairing Must Contract Opposite Variances

The evaluation operation only produces a basis-independent scalar because one argument transforms contravariantly and the other transforms covariantly. Pairing two vectors, or two covectors, directly by multiplying components does not yield a basis-independent number unless additional structure, such as a metric tensor, is introduced to convert one type into the other first.

Diagrammatic Summary

The diagram shows a covector f drawn from V* and a vector v drawn from V entering the evaluation operation together, with the result collapsing to a single scalar in the field F, reflecting the fact that all indices are consumed by the contraction and none remain free.