3.12.1 Tensor Double Dual Carrier Structure

Tensor Double Dual Carrier Structure is the examination of V** as a vector space in its own right, focusing on the vector space operations it carries as the dual of V*, and on how the canonical embedding ι : V -> V** faithfully carries the vector space structure of V itself into this newly constructed space. Before V** can be meaningfully identified with V, it must first be verified that V** is a legitimate vector space and that ι respects every operation that defines vector space structure, rather than merely matching elements in a set-theoretic sense.

The Vector Space Structure Carried by V**

Addition of Elements of V**

An element of V** is a linear functional on V*. Addition of two such functionals φ, ψ in V** is defined pointwise, exactly as addition is defined on any space of functions into a field:

for every f in V*. This is the same pattern already used to define addition on V* itself, applied one level higher.

Scalar Multiplication and the Zero Element

Scalar multiplication is likewise pointwise, (cφ)(f) = c \cdot φ(f), and the zero element of V** is the functional sending every covector to 0. These operations satisfy the standard vector space axioms because they are inherited directly from the field structure of F, exactly as for any space of functions into F.

Verifying That ι Carries Structure Faithfully

Additivity of the Embedding

For the identification of V with its image inside V** to be meaningful as vector spaces, not merely as sets, the embedding ι must respect addition:

using the linearity of f itself and the pointwise definition of addition in V**. This confirms ι(v_1 + v_2) = ι(v_1) + ι(v_2).

Homogeneity of the Embedding

A parallel computation using homogeneity of f confirms ι(cv) = c \cdot ι(v), so ι is fully linear, carrying both vector space operations of V faithfully into the corresponding operations of V**.

Why Carrying Structure, Not Just Elements, Matters

Set-Theoretic Correspondence Would Not Suffice

A mere bijection of underlying sets between V and V** would not justify treating them as the same vector space; addition and scalar multiplication on one side would need to be translated arbitrarily to make sense on the other, with no guarantee of consistency. The fact that ι is linear ensures that the entire algebraic structure, not just individual elements, transfers correctly.

Consistency with the Natural Pairing

The structure-carrying property of ι is exactly what makes expressions like <v_1 + v_2, f> = <v_1, f> + <v_2, f> valid regardless of whether the left-hand side is interpreted as ordinary evaluation of f on a sum of vectors in V, or as evaluation of the sum ι(v_1) + ι(v_2) on f from the V** side; both interpretations agree precisely because ι respects addition.

The Carrier Structure Under the Canonical Isomorphism

Basis Correspondence

In finite dimensions, since ι is an isomorphism, it carries a basis e_1, ..., e_n of V to a basis ι(e_1), ..., ι(e_n) of V**, and this image basis coincides exactly with the double-dual basis obtained by taking the dual of the dual basis e^1, ..., e^n, closing the loop between the original basis and its iterated dual construction.

Preserving Subspace Structure

Because ι is a linear isomorphism in finite dimensions, it also carries subspaces of V to corresponding subspaces of V** of the same dimension, and linear maps on V correspond, under this identification, to linear maps on V** acting compatibly, preserving the full lattice of subspace and mapping structure between the two spaces.

Diagrammatic Summary

The diagram illustrates that adding vectors in V before applying ι produces the same result as applying ι first and adding in V**, confirming the embedding carries additive structure faithfully.