dualitetspairing

Dualitetspairing, or duality pairing, is a bilinear map that relates two mathematical structures so that each can be recovered from the other through a duality. In the most common setting, it is a bilinear form P: A × B → S, where A and B are vector spaces or abelian groups and S is a field or abelian group (often a field F or the circle group). The pairing is written as ⟨a, b⟩ or P(a, b).

A central example in linear algebra is the canonical evaluation pairing between a vector space V and its dual V*: ⟨v, φ⟩ = φ(v). This pairing is nondegenerate: if ⟨v, φ⟩ = 0 for all φ, then v = 0, and similarly for φ. In finite dimensions this nondegeneracy yields a natural isomorphism V ≅ V**, making the duality between a space and its dual intrinsic. In infinite dimensions, the pairing can fail to be perfect, and one must distinguish V* from the full bidual.

Beyond vector spaces, duality pairings appear in many contexts. For instance, the dot product is a symmetric pairing on R^n, while a symplectic form is an alternating nondegenerate pairing on a vector space. In abstract harmonic analysis, Pontryagin duality uses a natural evaluation pairing between a locally compact abelian group G and its dual Ĝ, leading to reflexivity under suitable conditions. In algebraic geometry and topology, various duality theories produce pairing maps between cohomology groups that reveal deep structural information.

The term dualitetspairing mirrors the English concept of a duality pairing and denotes the same idea: a bilinear, often nondegenerate, relation that identifies a space with a dual object and facilitates the transfer of structure between them.