3.3.2 Tensor Linear Functional Codomain Field

Tensor Linear Functional Codomain Field is the fact that a linear functional's codomain is always the scalar field F underlying the vector space V, together with the structural consequences of this fixed, one-dimensional codomain: the surjectivity of every nonzero functional, the field-dependence of what "linear" even means, and how the specific choice of field, real, complex, or otherwise, affects which functionals exist and how they behave.

The Codomain Is Always F, Never V Itself

Fixed by Definition

A linear functional on V is, by definition, a map ω : V → F, with codomain always the scalar field, never V itself or any other vector space; this is what distinguishes a functional from a general linear map φ : V → W, discussed in tensor linear map codomain space, where W can be any vector space, including V itself in the special case of an endomorphism.

The Field as a One-Dimensional Vector Space Over Itself

F is regarded as a vector space over itself for the purposes of this definition, one-dimensional with any nonzero scalar, typically 1, serving as a basis; this is what makes ω : V → F a special case of a general linear map with a specific, fixed one-dimensional codomain, rather than an unrelated notion requiring separate development.

Surjectivity of Nonzero Functionals

Every Nonzero Functional Hits Every Scalar

If ω ∈ V* is nonzero, then ω is surjective onto F: given any v_0 with ω(v_0) = a ≠ 0, and any target scalar c, the vector (c/a) v_0 satisfies ω((c/a) v_0) = (c/a) ω(v_0) = c. This surjectivity is what guarantees, as used in tensor linear functional structure, that every level set {v : ω(v) = c} of a nonzero functional is nonempty for every c ∈ F, not merely for c = 0.

The Only Non-Surjective Functional Is Zero

Because F has no proper nonzero subspaces, other than {0}, the image of any functional ω : V → F is either {0}, when ω is the zero functional, or all of F, when ω is nonzero; there is no intermediate possibility, unlike a general linear map into a higher-dimensional codomain W, whose image can be any subspace of W of any intermediate dimension.

Field-Dependence of Functional Behavior

Real Versus Complex Codomain

When F = ℝ, the codomain of a functional carries an ordering, so real-valued functionals support notions such as positivity, ω(v) > 0, and half-spaces, {v : ω(v) ≥ 0}, that have no direct analogue when F = ℂ, since the complex numbers are not ordered; complex-valued functionals instead naturally pair with conjugate-linear structures in contexts requiring an inner-product-like pairing, a distinction absent from the real case.

Field Characteristic and Degenerate Behavior

Over a field of positive characteristic p, certain constructions built from functionals, such as symmetrization formulas involving division by k! for defining Sym^k(V*) projections, can fail to be well defined once p divides the relevant factorial; the codomain field's characteristic is therefore not a detail external to functional structure but a factor that can determine which auxiliary operations, beyond the basic linear structure itself, remain valid.

Consistency With General Tensor Codomain Conventions

Matching the Scalar Codomain of Multilinear Functionals

The fixed scalar codomain of a single covector is the base case of the general convention, established in tensor dual space structure, that every (p, q)-tensor, regarded as a multilinear functional, is ultimately valued in F once all of its vector and covector arguments have been supplied; a covector supplies this scalar output after consuming exactly one vector argument, the simplest nontrivial instance of the pattern followed by every higher tensor.

Why the Codomain Field Cannot Vary Within a Single Tensor Construction

Every factor of a tensor space built from V and V*, whether contributing an upper or a lower index, is defined over the same fixed field F as V itself; mixing codomains, for instance combining a functional valued in one field with a vector space defined over a different field, falls outside the scope of the tensor and dual-space constructions described throughout this material, which presuppose a single, fixed scalar field common to every space involved.