3.13.1 Tensor Vector Double Dual Assignment

Tensor Vector Double Dual Assignment is the specific rule that assigns to each vector v in a vector space V a corresponding element ev(v) of the double dual space V**, defined by letting ev(v) act on any covector φ through the evaluation ev(v)(φ) = φ(v). This assignment is the concrete, pointwise content of the canonical map between V and V**: rather than merely asserting that such a map exists, the assignment specifies exactly which element of V** corresponds to a given vector v, and how that element behaves when fed an arbitrary covector as input.

The Assignment Rule

Statement of the Rule

For a vector space V over a field F, with dual space V* and double dual V**, the assignment sends each v in V to the functional ev(v): V* → F given by:

for every φ in V*. The assignment fixes v and lets φ vary, in contrast to the original functional φ itself, which fixes φ and lets v vary. This role-reversal is the essential mechanism of the assignment: the vector v, ordinarily an input to covectors, is repackaged as an object that takes covectors as its own input.

Reading the Assignment as "Plugging In"

A convenient way to describe the assignment is that ev(v) is the operation of "plugging v in" to whatever covector is supplied. Every covector φ is a rule for producing a scalar from a vector; the assignment ev(v) reverses the direction of description, producing a scalar from a covector by feeding it the fixed vector v. The scalar output is identical in both directions, since ev(v)(φ) and φ(v) are defined to be equal.

Verifying the Assignment Lands in the Double Dual

Linearity in the Covector Argument

For ev(v) to qualify as a genuine element of V**, it must be linear as a function of φ. This is checked directly using the vector space structure of V*, in which addition and scalar multiplication of covectors are defined pointwise:

for scalars a, b and covectors φ, ψ. Since this equality holds for every choice of φ and ψ, ev(v) is confirmed to be a linear functional on V*, and therefore a legitimate member of V**.

Independence from Any Basis Choice

The assignment v ↦ ev(v) is defined purely in terms of the pairing between V and V*, without reference to a basis of either space. This distinguishes the assignment from other maps between V and dual-type spaces, such as the isomorphism from V to V* obtained by matching a chosen basis to its dual basis, which changes if a different basis is selected. The double dual assignment produces the same functional ev(v) regardless of what basis, if any, is used to compute it.

Linearity of the Assignment Itself

Additivity and Homogeneity in v

Beyond the linearity of each individual ev(v) in its covector argument, the assignment v ↦ ev(v) is itself linear as a function of v. For vectors v, w and scalars a, b:

This holds because, for any covector φ, both sides evaluated at φ reduce to a·φ(v) + b·φ(w), using the linearity of φ on V. The assignment is therefore a linear map ev: V → V**, not merely a set-theoretic assignment of elements.

Behavior on the Zero Vector

The assignment sends the zero vector of V to the zero functional of V**, since ev(0)(φ) = φ(0) = 0 for every covector φ. This is consistent with the linearity of the assignment, as any linear map must send the zero vector of its domain to the zero vector of its codomain.

Injectivity of the Assignment on Finite-Dimensional Spaces

Distinguishing Vectors by Their Covector Values

The assignment is injective on a finite-dimensional space V, meaning distinct vectors are always assigned to distinct elements of V**. This follows from the fact that if v is a nonzero vector, some covector φ can be found with φ(v) ≠ 0. Such a φ can be constructed by extending v to a basis of V and taking the corresponding dual basis functional that reads off the coefficient of v.

Consequence for the Assigned Functional

If φ(v) ≠ 0 for some φ, then ev(v)(φ) ≠ 0, so ev(v) cannot be the zero functional, and in particular ev(v) ≠ ev(0). Applying this to the difference of two vectors shows that ev(v) = ev(w) forces v = w, establishing that the assignment never merges two distinct vectors into the same double-dual element.

Component Description of the Assignment

Action on a Dual Basis

Given a basis e_1, ..., e_n of V with dual basis e^1, ..., e^n of V*, the assignment of a basis vector e_i acts on the dual basis functionals as:

where δ^j_i is the Kronecker delta, equal to 1 when i = j and 0 otherwise. This shows that ev(e_i) behaves, on the dual basis, exactly like the double dual basis element that "picks out" the i-th coordinate, which is precisely the vector one expects e_i to correspond to under a faithful identification of V with V**.

Coordinate Formula for a General Vector

For a general vector v = v^1 e_1 + ... + v^n e_n, the assigned functional ev(v) acts on a general covector φ = φ_1 e^1 + ... + φ_n e^n by:

so the components of ev(v) with respect to the double dual basis induced by e^1, ..., e^n are numerically identical to the components v^1, ..., v^n of v itself, matching the intuitive expectation that the assignment simply carries v's coordinates over unchanged.

Diagrammatic Summary

The diagram traces the assignment starting from a vector v, passing through the functional ev(v) that it becomes in V**, and ending at the scalar φ(v) produced once ev(v) is applied to an arbitrary covector φ. The output value at the far right is what confirms the assignment is faithful to the original pairing between V and V*.