3.12.5 Tensor Double Dual Reflexive Context

Tensor Double Dual Reflexive Context is the theoretical setting in which a finite-dimensional vector space is identified with the dual of its own dual space through a canonical, basis-independent isomorphism, establishing that the double dual construction returns, in essence, the original space rather than a new one. Given a vector space V over a field F, its dual space V* consists of all linear functionals on V, and the double dual V** consists of all linear functionals on V*. The reflexive context is the framework describing the natural map V → V** and the precise conditions, namely finite dimensionality, under which this map is an isomorphism, allowing tensor constructions to treat V and V** as the same object without loss of information.

The Canonical Evaluation Map

Definition of the Map

The bridge between V and V** is not an arbitrary isomorphism but a specific, canonically defined map ev: V → V** called the evaluation map. For each vector v in V, the map assigns an element ev(v) of V**, which is itself a functional acting on V*. This functional is defined by evaluating any covector φ in V* at the vector v:

for every φ in V*. The vector v is thereby repackaged as a rule that takes a covector as input and returns a scalar, which is exactly the type of object that lives in V**.

Why the Map Is Well Defined

For ev(v) to be a legitimate element of V**, it must itself be linear as a function of φ. This holds automatically because the vector space operations on V* are defined pointwise: for covectors φ and ψ and scalars a, b, the combination (aφ + bψ)(v) equals a·φ(v) + b·ψ(v) by the definition of addition and scalar multiplication in V*. Consequently ev(v) is linear in φ, confirming that ev(v) belongs to V** for every choice of v.

Linearity and Injectivity of ev

The map ev itself is linear in v, since ev(av + bw)(φ) = φ(av + bw) = aφ(v) + bφ(w) = a·ev(v)(φ) + b·ev(w)(φ) for all φ. If V is finite-dimensional, ev is also injective: if v is nonzero, there exists a covector φ with φ(v) ≠ 0, guaranteed by extending v to a basis and using the corresponding dual basis functional, so ev(v) cannot be the zero functional.

Reflexivity in Finite Dimensions

The Dimension Count Argument

When V has finite dimension n, the dual space V* also has dimension n, since a basis of V induces a dual basis of V* with the same number of elements. Applying this fact twice shows that V** also has dimension n. Since ev: V → V** is a linear injective map between spaces of equal finite dimension, it must also be surjective, and therefore an isomorphism.

The Meaning of "Canonical"

The isomorphism ev is called canonical or natural because its definition requires no arbitrary choice of basis. This distinguishes it from the isomorphism between V and V*, which does exist in finite dimensions but depends on choosing a basis of V to construct the corresponding dual basis. Because ev is basis-independent, V and V** are considered the same space in a structural sense, and identifying v with ev(v) is regarded as harmless.

Reflexive Spaces

A vector space for which the evaluation map ev is an isomorphism is called reflexive. Every finite-dimensional vector space is reflexive. This reflexive property is the technical content behind the double dual reflexive context: it certifies that dualizing twice returns a space that can be canonically identified with the space one started with.

Failure of Reflexivity in Infinite Dimensions

Where the Argument Breaks

The dimension-counting argument that guarantees surjectivity relies on V being finite-dimensional. For an infinite-dimensional vector space, the dual space V* typically has strictly larger dimension than V, so the evaluation map ev: V → V** remains injective but fails to be surjective. The image of ev in this case is a proper subspace of V**, consisting only of those functionals on V* that arise from evaluation at some vector of V.

Consequence for Tensor Constructions

This failure is the reason the reflexive context explicitly restricts attention to finite-dimensional spaces before treating V and V** interchangeably in tensor algebra. Constructions such as identifying a (p, q)-tensor built from copies of V and V* with a multilinear map on the corresponding dual spaces implicitly invoke the isomorphism V ≅ V**; in the infinite-dimensional setting this identification is not automatically available and must be handled with additional care or abandoned.

Role Within Tensor Algebra

Reinterpreting Vectors as Functionals on Covectors

Once the reflexive identification V ≅ V** is accepted, a vector v in V may be treated as a linear functional on V*, meaning it accepts a covector and returns a scalar. This reinterpretation is what allows a (1, 0)-tensor, ordinarily understood as an element of V, to also be described as a multilinear map taking one covector argument, matching the multilinear-map perspective used throughout tensor algebra.

Consistency of the Two Tensor Perspectives

The double dual reflexive context is what guarantees that the tensor product perspective, in which a (p, q)-tensor is an element of p copies of V tensored with q copies of V*, agrees with the multilinear map perspective, in which the same tensor is viewed as a multilinear map from p copies of V* and q copies of V to the field. Without the identification V ≅ V**, the first perspective would only naturally produce maps out of V** rather than out of V itself, and the two descriptions would not coincide.

Compatibility with Change of Basis

Because ev does not depend on a choice of basis, the identification of V with V** is preserved under any change of basis applied to V. This is essential for tensors, whose defining property is a specific transformation behavior under change of basis: an identification that itself depended on a basis choice would interfere with, rather than support, the basis-independent transformation law that characterizes a tensor as a well-defined geometric object.

Diagrammatic Summary

The diagram traces V through one application of the dual construction to reach V*, and through a second application to reach V**. The dashed arrow labeled ev represents the direct, basis-independent path from V to V**, which is an isomorphism precisely in the reflexive, finite-dimensional context.