3.13.4 Tensor Canonical Embedding Injectivity

Tensor Canonical Embedding Injectivity is the property of the canonical map ev: V → V** guaranteeing that distinct vectors of V are always sent to distinct functionals in V**, so that no information about V is lost or collapsed when it is embedded into its double dual. Injectivity is what turns ev into a faithful copy of V inside V**, and it is the property that, combined with a dimension count, upgrades the embedding into a full isomorphism in the finite-dimensional case.

Statement of Injectivity

The Formal Condition

A linear map is injective exactly when its kernel contains only the zero vector. Applied to the canonical embedding, injectivity of ev means:

Because ev is already known to be linear, this single kernel condition is equivalent to the more general statement that ev(v) = ev(w) forces v = w for any two vectors v, w in V, since ev(v) = ev(w) implies ev(v - w) = 0 by linearity, which by the kernel condition forces v - w = 0.

Restatement in Terms of the Evaluation Pairing

Unwinding the definition of ev, the condition ev(v) = 0 means that the functional ev(v) sends every covector φ to 0, that is, φ(v) = 0 for every φ in V*. Injectivity of ev therefore restates as: the only vector annihilated by every covector in V* is the zero vector.

Proof of Injectivity in Finite Dimensions

Setting Up the Argument

Suppose V is finite-dimensional and v is a nonzero vector. To show ev(v) ≠ 0, it suffices to exhibit a single covector φ with φ(v) ≠ 0, since this shows ev(v) does not annihilate every covector and is therefore not the zero functional.

Constructing the Detecting Covector

Since v is nonzero, it can be extended to a basis v = e_1, e_2, ..., e_n of V. Let e^1, ..., e^n denote the corresponding dual basis of V*, characterized by:

Taking φ = e^1 gives φ(v) = e^1(e_1) = 1 ≠ 0, which is exactly the required detecting covector. Since such a φ exists for every nonzero v, the kernel of ev contains no nonzero vectors, establishing injectivity.

Why Finite Dimensionality Matters Here

The construction relies on being able to extend v to a finite basis of V and form a well-defined dual basis, both of which are standard facts of finite-dimensional linear algebra. In the infinite-dimensional setting a basis still exists in the sense of a Hamel basis, and an analogous detecting functional can still typically be constructed, so injectivity of ev in fact persists in the infinite-dimensional case as well; it is surjectivity, not injectivity, that is the property lost when V fails to be finite-dimensional.

Geometric Meaning of Injectivity

No Collapsing of Vectors

Injectivity means the embedding never identifies two genuinely different vectors with a single element of V**. If v ≠ w, their images ev(v) and ev(w) remain distinguishable functionals on V*, meaning some covector can always be found that evaluates differently on v than on w.

The Image as a Faithful Copy

Because ev is both linear and injective, its image ev(V) inside V** is a subspace isomorphic to V itself, via the restriction of ev to a bijection onto its image. This image is described as a faithful copy of V: every algebraic relation that holds among vectors of V, such as linear dependence or independence, holds identically among their images in ev(V).

Injectivity as Half of the Reflexive Isomorphism

Combining with Dimension Counting

In the finite-dimensional case, injectivity of ev combined with the equality of dimensions:

forces ev to also be surjective, by the rank–nullity theorem applied to a linear map between finite-dimensional spaces of equal dimension with trivial kernel. Injectivity is therefore the more fundamental of the two properties: it holds under weaker hypotheses and, together with a dimension count that is available only in finite dimensions, yields the stronger conclusion of surjectivity.

Injectivity Without Surjectivity

When V is infinite-dimensional, injectivity of ev still holds, but the dimension-counting argument for surjectivity breaks down, since V** generally has strictly larger dimension than V in that setting. The embedding remains a faithful, information-preserving copy of V, but it no longer accounts for the entirety of V**.

Diagrammatic Summary

The diagram shows two distinct starting points v and w mapped by the injective embedding to two distinct endpoints ev(v) and ev(w), with no crossing of arrows into a shared image point, illustrating that injectivity forbids any collapsing of the domain when passing into V**.