Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Linear functionals and the algebraic dual
Definition
Let be a vector space over . A linear functional on is a linear map , where is regarded as a vector space over itself. The algebraic dual space of is
with pointwise addition and scalar multiplication. This is the full algebraic dual: no topology, norm, or continuity condition is imposed.
Depends on
Used by
- The annihilator U^∘≤ V^* of U≤ V and the preannihilator ^∘ S≤ V of S≤ V^* Definition
- The canonical evaluation map J_V:V→ V^** given by J_V(v)(f)=f(v) Definition
- The dual family (b^*)_b∈ B associated to a Hamel basis B, defined by b^*(c)=δ_bc Definition
- The transpose or algebraic adjoint T^*:W^*→ V^*, T^*(g)=g∘ T, of a linear map T:V→ W Definition
- Assuming choice, if v∉ U≤ V, some f∈ V^* vanishes on U and satisfies f(v)=1 Lemma
- Bilinear forms on V correspond linearly and bijectively to linear maps V→ V^* Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 10 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- H. Pinkham, Linear Algebra, §6.1 (standard reference, not scraped)
- K. Conrad, Infinite-Dimensional Dual Spaces (standard reference, not scraped)