3.13 Tensor Canonical Double Dual Embedding Structure

Tensor Canonical Double Dual Embedding Structure is the overarching algebraic structure formed by the natural, basis-independent map that embeds a vector space V into the double dual space V**, together with all of the properties that make this embedding well behaved: linearity, injectivity, and, in the finite-dimensional case, surjectivity. This structure is the general framework from which the specific evaluation map and its assignment rule are drawn, and it organizes the relationship between V and V** as a single coherent piece of algebra rather than a collection of isolated facts.

The Embedding as a Structure

What "Structure" Refers To

The word structure here refers to the full package of data associated with the map ev: V → V**: its defining formula, its algebraic properties as a linear map, its behavior with respect to bases, and the distinction between the injective embedding that always exists and the isomorphism that exists only under finite dimensionality. Presenting these together as a structure clarifies which facts hold unconditionally and which depend on extra hypotheses.

The Defining Map

At the core of the structure is the evaluation map, which sends each vector v in V to the functional ev(v) in V** defined by:

for every covector φ in V*. This single formula generates the entire embedding structure: all of its properties are derived consequences of this definition rather than separate postulates.

Universality Across Dimension

The Embedding Always Exists

Unlike the isomorphism between V and V*, which requires choosing a basis, and unlike the isomorphism between V and V**, which requires finite dimensionality, the map ev is defined for every vector space, regardless of dimension, and requires no choice of basis to write down. This universality is a central feature of the structure: it is available as a starting point in any setting, with its strength depending afterward on further hypotheses about V.

Injectivity as a Universal Property

The embedding ev is injective whenever V has enough covectors to separate its points, a property guaranteed by the Hahn–Banach-type separation available in finite dimensions and, more generally, whenever the dual space is large enough to distinguish any two distinct vectors. In the finite-dimensional case, injectivity holds unconditionally, since a nonzero vector can always be extended to a basis and paired with a dual basis functional that detects it.

Surjectivity as the Dimension-Dependent Property

Surjectivity of ev, in contrast, is not universal. It holds when V is finite-dimensional, because then V, V*, and V** all share the same finite dimension, forcing the injective map ev to also be onto. When V is infinite-dimensional, V* is generally of strictly larger dimension than V, so V** is larger still, and ev embeds V as a proper subspace of V** without filling it entirely.

Algebraic Properties of the Embedding

Linearity

The map ev is linear: for vectors v, w and scalars a, b,

This makes the embedding structure compatible with all of the standard operations of linear algebra: sums, scalar multiples, and by extension linear combinations, spans, and subspaces are respected by the passage from V into V**.

Compatibility with Linear Maps

If T: V → W is a linear map between two vector spaces, the embedding structure interacts with T through a corresponding map T**: V** → W**, obtained by dualizing T twice. The embeddings of V and W are compatible with this construction, in the sense that applying T first and then embedding into W** gives the same result as embedding into V** first and then applying T**. This compatibility is what allows the embedding to be treated as a natural transformation rather than a map defined space-by-space in isolation.

Preservation of Subspace Relations

If U is a subspace of V, the embedding structure carries U into a corresponding subspace of V** in a manner consistent with the embedding of V itself, so that the containment relation U ⊆ V is mirrored by a containment relation between the embedded images. This preservation of subspace structure is part of what justifies treating the embedding as respecting the full algebraic shape of V, not merely its underlying set of vectors.

The Embedding Structure and the Reflexive Case

Isomorphism as a Special Case

When the embedding happens to be surjective, meaning V is finite-dimensional, the general embedding structure specializes to the reflexive case, in which ev is not just an embedding but a full isomorphism V ≅ V**. The embedding structure is therefore the broader framework, and the reflexive identification is the particular instance of it that occurs under the finite-dimensionality hypothesis.

Structure Retained Even Without Surjectivity

Even when V is infinite-dimensional and ev fails to be surjective, the embedding structure retains its full content as an injective linear map that respects the algebraic operations on V and interacts naturally with linear maps between vector spaces. The failure of surjectivity removes only the identification of V** with V; it does not disturb any of the other properties of the embedding.

Diagrammatic Summary

The diagram shows the embedded image ev(V) sitting inside V**, drawn smaller than the surrounding rectangle to emphasize that it need not fill the entire space. The embedding structure guarantees ev(V) is always a faithful linear copy of V, and the finite-dimensional case is the special circumstance in which this copy expands to occupy all of V**.