3.2.4 Tensor Covector Dual Space Membership

Tensor Covector Dual Space Membership is the precise criterion by which a function V → F is recognized as belonging to V*, namely satisfying both linearity conditions, additivity and homogeneity, and the practical consequences of this membership test: how it is verified for a candidate function, how it is used to confirm that specific familiar constructions, such as coordinate-reading functionals and directional-derivative functionals, do or do not qualify, and how membership in V* differs from membership in the larger set of all functions from V to F.

The Membership Criterion

Statement of the Test

A function f : V → F belongs to V* if and only if it satisfies both:

for all v_1, v_2 ∈ V, and f(cv) = c f(v) for all v ∈ V and scalars c. Both conditions must hold for every input, not merely for some special vectors; a single counterexample to either condition disqualifies f from membership in V*.

Sufficiency of the Two Conditions

No further conditions are needed: the two properties, additivity and homogeneity, are jointly sufficient to guarantee that f behaves consistently with the full vector-space structure of V, including compatibility with arbitrary finite linear combinations, f(Σ c_i v_i) = Σ c_i f(v_i), which follows by repeated application of additivity and homogeneity rather than requiring separate verification.

Examples That Satisfy Membership

Coordinate-Reading Functionals

For a fixed basis {e_i} of V, the function e^j(v) = v^j, reading off the j-th coordinate of v = Σ v^i e_i, satisfies both conditions: e^j(v_1 + v_2) = (v_1 + v_2)^j = v_1^j + v_2^j = e^j(v_1) + e^j(v_2), and e^j(cv) = (cv)^j = c v^j = c e^j(v), since coordinates themselves add and scale linearly under vector addition and scalar multiplication. This confirms directly, from the membership test, that the dual basis vectors e^j are legitimate elements of V*.

Zero Functional

The constant-zero function f(v) = 0 for all v trivially satisfies both conditions, 0 + 0 = 0 and c · 0 = 0, and is the zero element of V*, the additive identity described in dual spaces and covectors.

Examples That Fail Membership

A Function Failing Additivity

The function f(v) = ‖v‖, the Euclidean norm, generally fails additivity, since the triangle inequality ‖v_1 + v_2‖ ≤ ‖v_1‖ + ‖v_2‖ is typically strict rather than an equality, so f(v_1 + v_2) ≠ f(v_1) + f(v_2) for most choices of v_1 and v_2; the norm is not a covector, despite being a perfectly well-defined function V → F when F is the reals.

A Function Failing Homogeneity

The function f(v) = (v^1)^2, squaring the first coordinate, fails homogeneity, since f(cv) = (cv^1)^2 = c^2 (v^1)^2 = c^2 f(v), which equals c f(v) only when c ∈ {0, 1}; this quadratic dependence on the scalar c disqualifies f from V*. The function also fails additivity, since (v_1^1 + v_2^1)^2 ≠ (v_1^1)^2 + (v_2^1)^2 in general, so f is excluded from V* on both grounds independently.

Membership Relative to the Larger Function Space

V* as a Proper Subset of All Functions V → F

The set of all functions from V to F is, for infinite F, generally a much larger space than V*, and membership in V* is a strict, nontrivial restriction; V* singles out exactly those functions compatible with the linear structure of V, discarding every function exhibiting any nonlinear dependence on its input.

Membership Is Preserved by the Dual-Space Operations

If f, g ∈ V*, then f + g and cf also satisfy the membership test, ((f+g)(v_1+v_2) = f(v_1+v_2)+g(v_1+v_2) = f(v_1)+f(v_2)+g(v_1)+g(v_2), matching (f+g)(v_1)+(f+g)(v_2), and similarly for homogeneity, confirming that the membership criterion is closed under the vector-space operations already defined on V*; this closure is precisely what makes V* a well-defined subspace of the space of all functions V → F, rather than merely an unstructured collection of individually linear functions.