3.5.2 Tensor Basis Covector Evaluation Target

Tensor Basis Covector Evaluation Target is the precise characterization of what a basis covector e^i is legitimately permitted to act on, namely any vector belonging to V, the same domain the entire dual basis was constructed relative to, together with the consequences of this restriction: that basis covectors cannot be meaningfully applied to vectors from an unrelated space, cannot be applied to covectors, and produce well-defined, unambiguous scalar output for every legitimate target without exception.

The Legitimate Target Set

Any Vector in V, Without Restriction Beyond Membership

A basis covector e^i, defined relative to a basis {e_j} of V, accepts as its evaluation target any vector v ∈ V whatsoever, not merely the basis vectors e_j used in its construction; the extension by linearity from e^i(e_j) = δ^i_j to e^i(v) = v^i for a general v, established in tensor dual basis evaluation rule, guarantees that every element of V, without exception, is a valid target.

Well-Definedness for Every Target

Because e^i is a total function on V, every legitimate target produces exactly one scalar output, v^i, with no target vector left unevaluated and no target producing more than one possible output; this totality is inherited directly from e^i's status as an ordinary linear functional, e^i ∈ V*, and requires no special argument beyond the general theory of linear functionals already established.

Illegitimate Targets

Vectors From an Unrelated Space

A vector u belonging to a different vector space U, not equal to V and not identified with V via any specified map, is not a legitimate target for e^i; the expression e^i(u) is simply undefined, since e^i's domain is fixed at V specifically, in accordance with the covector domain relation established elsewhere. Supplying such a u is a type mismatch, not a computation that merely evaluates to zero or produces some other degenerate but well-defined answer.

Covectors Themselves

Because e^i : V → F and not e^i : V* → F, a covector ω ∈ V* is likewise not a legitimate target for e^i; attempting to evaluate e^i(ω) conflates the roles of vector and covector, treating e^i as though it accepted arguments from V*, which it does not. The natural pairing that does make sense between e^i and a covector runs in the opposite direction, evaluating ω(e_i) instead, using e_i as the vector-side input to ω.

Coordinates as the Standard Description of a Target

Target Description Relative to the Same Basis

When a target v is described using the coordinates v = Σ_j v^j e_j relative to the very basis {e_j} that produced e^i, evaluation reduces immediately to e^i(v) = v^i, requiring no further computation beyond reading off the appropriate coefficient. This is the most direct and computationally convenient way a target can be presented for evaluation against a basis covector.

Target Described Relative to a Different Basis

If a target v is instead given in coordinates relative to a different basis {e'_j}, evaluation still succeeds, since v remains a legitimate element of V regardless of how it happens to be described, but computing e^i(v) requires first converting v's coordinates into the {e_j} basis, using the transition matrix between the two bases, before the direct coordinate-reading shortcut e^i(v) = v^i can be applied.

Consistency With Slot Role in Higher Tensors

The Vector-Accepting Slot of a Multilinear Functional

The evaluation-target restriction for a single basis covector is the elementary instance of the general fact, described in tensor covector tensor slot role, that every lower-index slot of a (p, q)-tensor accepts only vectors from the specific domain V the tensor was built from; the legitimate-target analysis for e^i generalizes directly, slot by slot, to every lower-index argument position of any higher-rank tensor constructed from covectors on V.