duals

Duals are objects or structures that correspond to a dual notion under a formal correspondence, often reversing roles of elements and functions or exchanging operations with arguments. The idea of duality appears across mathematics, computer science, and logic as a way to view problems from a complementary perspective.

In linear algebra, the dual space V* of a vector space V over a field F consists of all linear functionals from V to F. The evaluation pairing allows every v in V to define a functional on V*, and conversely each functional in V* evaluates on V. When V is finite dimensional, dim V* equals dim V. For a linear map T: V -> W, its dual (or transpose) T*: W* -> V* is given by T*(g) = g ∘ T.

In geometry, duality takes several forms. For a convex polytope P in R^n, its dual P* comprises linear functionals bounded by 1 on P; faces of P correspond to faces of P* with complementary dimensions. In planar graphs, the dual G* has a vertex for each face of G and edges crossing original edges, a concept useful in embedding and planarity testing.

In coding theory, the dual code C⊥ consists of all vectors orthogonal to every codeword of C under a chosen inner product. In optimization, duality relates a primal problem to a dual problem, often via a Lagrangian, with weak and strong duality providing conditions for optimal solutions.

Beyond these, duality appears in category theory as contravariant equivalences and in physics as symmetries exchanging fundamental quantities. See also related topics such as dual spaces, dual graphs, dual polytopes, and dual codes.