3.12 Tensor Double Dual Space Structure

Tensor Double Dual Space Structure is the construction, properties, and significance of V**, the dual space of the dual space V*, obtained by applying the dual space construction a second time to a vector space V. Although formed by an entirely formal repetition of the same procedure used to build V* from V, the double dual carries a special, canonical relationship back to the original space V that does not hold between V and V* themselves, making V** one of the most structurally important constructions built from a single vector space and its dual.

Constructing the Double Dual

Definition

Given a vector space V over a field F, its dual V* is the space of linear functionals V -> F. Applying the same construction to V* gives its dual space, V** = (V*)*, consisting of all linear functionals V* -> F. An element of V** is therefore a rule that assigns a scalar to every covector in V*, subject to the usual requirements of linearity.

Iterating Further

The construction could in principle be repeated again to form V***, the dual of the double dual, but this iteration does not produce anything structurally new beyond V* and V**, since the canonical relationship established between V and V** extends naturally to identify V*** with V* itself, closing the pattern after just two steps.

The Canonical Embedding

Definition of the Embedding Map

There is a linear map ι : V -> V**, defined without any choice of basis, sending each vector v to the functional ι(v) on V* given by

for every covector f in V*. This map exists for every vector space, of any dimension, and requires only the natural pairing between V and V* to be defined.

Injectivity of the Embedding

The map ι is always injective: if ι(v) = 0, meaning f(v) = 0 for every covector f, then v must be the zero vector, since for any nonzero v a covector can always be constructed, using a basis extension argument, that assigns v a nonzero value.

Surjectivity and Finite Dimension

The Finite-Dimensional Case

When V is finite-dimensional, ι is also surjective, and hence a full linear isomorphism V ≅ V**. This can be verified directly by a dimension count: since dim(V*) = dim(V), applying the same equality again gives dim(V**) = dim(V*) = dim(V), and an injective linear map between spaces of equal finite dimension is automatically surjective.

The Canonical Nature of This Isomorphism

Unlike the isomorphism between V and V* itself, which always requires an arbitrary choice such as a basis, the isomorphism ι : V -> V** requires no such choice; it is built entirely from the natural pairing, making it canonical. This is the precise technical sense in which V and V** are considered "the same" space in finite dimensions, while V and V* are only ever isomorphic non-canonically.

Infinite-Dimensional Behavior: Reflexivity

Injective but Not Surjective

In infinite dimensions, ι remains injective but generally fails to be surjective, since dim(V**), obtained by applying the Erdős–Kaplansky dimension jump twice, is typically much larger than dim(V). A vector space for which ι is an isomorphism is called reflexive, and this property is the exception rather than the rule once infinite dimension is involved.

Reflexivity in Analysis

In functional analysis, where the topological dual is used instead of the full algebraic dual, reflexivity becomes a more delicate and interesting property: some infinite-dimensional spaces, such as Hilbert spaces, are reflexive with respect to their topological dual, while others are not, and this distinction plays a significant role in the theory of Banach spaces.

Consequences for Tensor Algebra

Identifying V with V** in Practice

Because of the canonical isomorphism in finite dimensions, tensor algebra freely identifies V with V**, treating a vector v and its associated functional ι(v) on V* as literally the same object without further comment, which is what licenses the symmetric pairing notation <v, f> = <f, v> used throughout discussions of the natural pairing.

The Double Dual as the Source of Symmetric Duality

The existence of a canonical V ≅ V** is what allows the relationship between V and V* to be treated as fully symmetric in finite dimensions, each acting as the dual of the other, even though the more direct identification V ≅ V* itself remains basis-dependent and non-canonical.

Diagrammatic Summary

The diagram shows the canonical, choice-free embedding ι of V into its double dual, which becomes a full isomorphism precisely when V is finite-dimensional.