3.6.1 Tensor Covector Vector Input

Tensor Covector Vector Input is the vector-side argument supplied to a covector when the evaluation operation ω(v) is carried out, examined specifically as the input half of that operation: what qualifies as a legitimate vector input, how the input's own coordinate description interacts with the mechanics of evaluation, and how the vector input, despite being consumed and reduced to a single scalar factor, retains its full linear structure throughout the evaluation process rather than being simplified or approximated in any way.

What Qualifies as a Vector Input

Membership in the Covector's Domain

The vector input v to ω(v) must belong to V, the specific domain the covector ω ∈ V* was defined relative to, matching the domain relation already established for covectors; no other requirement is imposed on v beyond this membership, so every element of V, without exception, is a valid input.

No Further Restriction Within V

Unlike some operations that behave specially on distinguished subsets, such as basis vectors or unit vectors, evaluation of a covector treats every vector input uniformly: there is no privileged class of "nicer" inputs that receive special treatment, and the linearity of ω guarantees identical algebraic behavior, ω(v_1 + v_2) = ω(v_1) + ω(v_2) and ω(cv) = cω(v), for every input without exception.

Coordinate Description of the Input

Reducing Evaluation to Coordinates

When v is presented in coordinates, v = Σ_j v^j e_j relative to some basis, and ω is presented in coordinates relative to the corresponding dual basis, ω = Σ_i ω_i e^i, evaluation reduces to the dot-product-like sum ω(v) = Σ_i ω_i v^i, established in tensor dual basis evaluation rule; the vector input's own coordinates v^i are exactly the quantities that get multiplied, term by term, against the covector's coordinates and summed.

The Input Need Not Match the Covector's Native Basis

The vector input can be presented in any basis, not necessarily the one used to define the covector's own coordinates; if a different basis is used for v, the coordinates of v must first be converted, via the appropriate transition matrix, into the basis matching ω's coordinates before the direct summation formula applies, or equivalently, ω's coordinates must be converted into the basis matching v's, since the final scalar result ω(v) is basis-independent regardless of which conversion route is taken.

Preservation of Linear Structure Through Evaluation

No Loss of Information About Linear Combinations

Because ω is linear, evaluating it on a vector input built as a linear combination of other vectors, v = c_1 u_1 + c_2 u_2, produces exactly c_1 ω(u_1) + c_2 ω(u_2), with no approximation, truncation, or loss of precision inherent to the operation itself; the full linear structure of how v relates to other vectors is faithfully reflected in how ω(v) relates to ω(u_1) and ω(u_2).

Vector Input as an Unaltered Argument

The vector input v itself is not modified, transformed, or consumed destructively by the evaluation operation; ω(v) produces a new scalar output while v remains available, unchanged, as an element of V for use in any other computation, exactly as calling an ordinary mathematical function on an argument leaves the argument itself intact.

Vector Input Within Multi-Slot Evaluation

One Argument Among Several for Higher Tensors

For a higher-rank tensor with multiple lower-index slots, each individual vector input occupies its own designated slot, as described in tensor covector tensor slot role, and the analysis of what qualifies as a legitimate input, membership in the correct domain space, applies independently to each slot's argument; a single covector's vector input is the simplest, one-slot instance of the general pattern governing vector arguments supplied to any multilinear tensor.

Consistency With the Natural Pairing

The vector input to a covector evaluation is exactly the second argument of the natural pairing V* × V → F described in dual spaces and covectors; framing evaluation as "supplying a vector input to a covector" and framing it as "the natural pairing applied to a covector-vector pair" are two descriptions of the identical operation, differing only in which of the two roles, functional or argument, is emphasized as the active one.