Dualprojection

Dualprojection, or dual projection, is a term used in several disciplines to describe a projection operation together with its dual counterpart defined through a duality between spaces. The exact meaning depends on context, but it typically involves an idempotent linear map and its action on a dual space or dual configuration.

In linear algebra and functional analysis, a projection P: V → V satisfies P^2 = P. The associated dual projection P: V → V is defined by P(f) = f ∘ P for f in the dual space V. P is also a projection: (P)^2 = P. Under a nondegenerate pairing, the subspace relations mirror the primal ones: range(P) equals the annihilator of ker(P), and ker(P) equals the annihilator of range(P). This dual projection provides a way to study subspaces of V by examining their annihilators in V.

In probability theory and Banach space theory, conditional expectation operators are canonical examples of projections on Lp spaces. Their duals act on the corresponding dual spaces Lq (1/p + 1/q = 1), functioning as dual projections. These duals preserve many structural properties and are important in areas such as martingale theory and ergodic analysis.

In projective geometry, a dual projection can refer to a projection interpreted in the dual projective space, where incidences between points and lines are exchanged under polarity or duality. Here, the concept adapts to mappings that respect the dual relationship between geometric objects.

Because dualprojection is context-dependent, it is used to describe the relationship between a projection and its dual under a given duality. See also projection, dual space, adjoint operator, and polarity.