Adjointtaking

Adjointtaking is a term used to describe the practice or operation of forming adjoints in mathematics. It encompasses both analytic constructions of adjoints of linear operators on inner product spaces and the categorical notion of adjoint functors, depending on the context.

In functional analysis and linear algebra, adjointtaking refers to forming the adjoint of a linear operator. If T is a bounded linear operator between Hilbert spaces H and K, there exists a unique operator T* such that ⟨Tx, y⟩ = ⟨x, T*y⟩ for all x in H and y in K. When H = K, T* is the adjoint of T. Key properties include (T*)* = T, (ST)* = T* S*, and ∥T*∥ = ∥T∥. In matrix terms, T* corresponds to the conjugate transpose under the standard inner product.

In category theory, adjointtaking also appears in the formation of adjoint functors. If F: C → D and G: D → C form an adjoint pair with F left adjoint to G (F ⊣ G), then adjointtaking refers to identifying or constructing this pair via the natural isomorphisms Hom_D(FX, Y) ≅ Hom_C(X, GY).

Examples and terminology: in practice, adjointtaking may be seen as taking the adjoint of an operator or identifying left/right adjoints in a functorial setting. The term adjointtaking is not widely standardized, but it serves as a compact label for the process of forming adjoints across contexts.

See also: adjoint, adjunction, dual space, transpose, Hermitian transpose.