3.12.2 Tensor Double Dual Element Structure

Tensor Double Dual Element Structure is the detailed anatomy of a single element of V**, examining what such an element fundamentally is, what conditions it must satisfy to qualify as a member of the double dual, and how, in finite dimensions, every such element is guaranteed to correspond to an ordinary vector of V through the canonical embedding. Understanding the internal structure of a double-dual element clarifies exactly why the identification V ≅ V** is legitimate rather than a notational convenience layered artificially on top of two unrelated spaces.

What an Element of V** Fundamentally Is

A Functional on Covectors

An element φ of V** is, by the defining construction of the double dual, a linear functional whose domain is V* and whose codomain is F. That is, φ takes a covector as input and produces a scalar, satisfying

for scalars a, b and covectors f_1, f_2 in V*. Any function on V* failing this linearity condition does not qualify as an element of V**, regardless of how it might otherwise be defined.

Distinguishing an Element of V** from an Element of V*

An element of V* consumes a vector from V and produces a scalar; an element of V** consumes a covector from V* and produces a scalar. Although both are, structurally, linear functionals, they operate on entirely different domains, and confusing the two, for example by attempting to apply an element of V** directly to a vector, is a category error unless the vector has first been converted into an element of V** via the canonical embedding.

Elements of V** Arising from the Canonical Embedding

The Form Taken by Embedded Elements

For a vector v in V, the corresponding element ι(v) of V** acts on a covector f by the rule ι(v)(f) = f(v). Every element of V** produced this way has the specific characteristic of being expressible as evaluation at a fixed point v, a property that does not hold for an arbitrary linear functional on an arbitrary vector space in general.

Finite-Dimensional Guarantee: Every Element Has This Form

In finite dimensions, because ι is surjective, every single element of V**, without exception, arises as ι(v) for some vector v in V. This means the internal structure of a double-dual element in finite dimensions is entirely predictable: it is always, in disguise, nothing more than evaluation at some particular vector, and no additional, more exotic functionals on V* exist beyond this evaluation type.

Coordinate Description of a Double-Dual Element

Components Relative to the Double-Dual Basis

Given a basis e_1, ..., e_n of V with dual basis e^1, ..., e^n, the double-dual basis of V** consists of the functionals E_1, ..., E_n dual to e^1, ..., e^n, satisfying E_i(e^j) = δ^j_i. Under the canonical identification, E_i corresponds exactly to ι(e_i), so an element φ of V** with components φ^i relative to this basis, φ = φ^i E_i, corresponds precisely to the vector v = φ^i e_i in V.

Consistency of the Coordinate Round Trip

Applying φ = φ^i E_i to the dual basis covector e^j gives φ(e^j) = φ^i δ^j_i = φ^j, confirming that the components of the double-dual element match, coordinate for coordinate, the components of the corresponding original vector, with no distortion introduced by passing through the double dual construction.

Elements of V** in Infinite Dimensions

Existence of Non-Evaluation Functionals

In infinite dimensions, since ι fails to be surjective, V** contains elements that cannot be expressed as evaluation at any vector of V. Such elements still satisfy the linearity condition required of any element of V**, but they lack the concrete, evaluation-based form that every double-dual element possesses in the finite-dimensional case.

Practical Impact

This means that in infinite dimensions, reasoning about "an arbitrary element of V**" must allow for the possibility of these non-evaluation functionals, and any proof or construction relying on every element of V** having the special form ι(v) must be restricted explicitly to the finite-dimensional setting, or otherwise justified through additional hypotheses such as reflexivity.

Diagrammatic Summary

The diagram shows a general element of V** as a linear functional on V*, which in the finite-dimensional case is always guaranteed to arise as the embedded image of a unique vector in V.