3.13.3 Tensor Canonical Embedding Linearity

Tensor Canonical Embedding Linearity is the property of the canonical embedding map ev: V → V** stating that it respects vector addition and scalar multiplication, meaning it is a linear map between vector spaces rather than merely an arbitrary assignment of elements. This linearity is what elevates the embedding from a set-theoretic injection into an algebraic one, ensuring that the copy of V sitting inside V** reproduces not just the individual vectors of V but the entire linear structure connecting them.

Statement of the Linearity Property

The Two Conditions

A map is linear when it satisfies additivity and homogeneity. For the canonical embedding, these two conditions read:

for all vectors v, w in V and all scalars a in the field F. Together these are usually combined into a single statement of linearity:

for arbitrary scalars a, b.

Proof of Additivity

Reducing to the Definition

To verify additivity, both sides of the equation ev(v + w) = ev(v) + ev(w) are elements of V**, meaning they are functionals on V*. Two functionals are equal precisely when they agree on every input, so it suffices to show that ev(v + w)(φ) = (ev(v) + ev(w))(φ) for an arbitrary covector φ.

Carrying Out the Computation

Starting from the left-hand side and applying the definition of ev:

Since φ is itself a linear functional on V, it satisfies φ(v + w) = φ(v) + φ(w). Continuing:

Since this holds for every φ, the functionals ev(v + w) and ev(v) + ev(w) are identical elements of V**, establishing additivity.

Proof of Homogeneity

Carrying Out the Computation

For homogeneity, the same strategy applies: evaluate both sides at an arbitrary φ. Starting from the left:

Since φ is linear, φ(av) = a·φ(v), so this becomes:

Again, since this holds for every φ, the functionals ev(av) and a·ev(v) coincide, establishing homogeneity.

Where Linearity of the Embedding Comes From

Two Layers of Linearity Involved

The proof of the embedding's linearity depends on the linearity of each individual covector φ as a functional on V. There are, in effect, two layers of linearity in play: the linearity of φ itself, which is used inside the computation, and the linearity of ev as a map on V, which is the conclusion being established. The first layer is what makes the second layer possible.

Consistency with the Vector Space Structure of V**

The linearity of ev also depends on V** being equipped with its own vector space structure, in which addition and scalar multiplication of functionals are defined pointwise. The proof implicitly uses this pointwise structure when it asserts that (ev(v) + ev(w))(φ) = ev(v)(φ) + ev(w)(φ), which is simply the definition of addition of functionals in V**.

Consequences of Linearity for the Embedding Structure

Preservation of Linear Combinations

Because ev is linear, it automatically preserves arbitrary finite linear combinations, not just sums of two vectors:

for scalars c_1, ..., c_k and vectors v_1, ..., v_k. This extension follows from repeated application of additivity and homogeneity, and it means that linear dependence relations among vectors in V are mirrored exactly by the corresponding relations among their images in V**.

A Prerequisite for Being an Isomorphism

Linearity is a necessary condition for ev to be an isomorphism in the finite-dimensional, reflexive case: an isomorphism of vector spaces must first be a linear map before the additional requirements of injectivity and surjectivity are even meaningful to check. The linearity established here is therefore the foundation on which the stronger reflexive identification of V with V** is built.

Preservation of the Zero Vector

As a direct corollary of linearity, ev sends the zero vector of V to the zero element of V**, since ev(0) = ev(0·0) = 0·ev(0) = 0, using homogeneity with scalar 0.

Diagrammatic Summary

The diagram shows that adding v and w before applying ev produces the same result as applying ev to each of v and w separately and then adding the outputs in V**, which is precisely the content of the additivity half of the embedding's linearity.