functionalsmappings

Functional mappings is a term used in mathematics to describe maps whose inputs are functions. More precisely, a functional is a map that takes a function from a function space and returns a scalar, typically a real or complex number. More generally, the broader idea of a mapping includes operators that take functions to other objects, such as functions or numbers, and may act between different function spaces.

Functionals are linear when they satisfy linearity: L(αf + βg) = αL(f) + βL(g) for all functions f and g in the domain and scalars α, β. Common examples include evaluation functionals, such as ev_x(f) = f(x), and integral functionals, such as L_g(f) = ∫ f(t) g(t) dt, where g is a fixed function. In Hilbert spaces, functionals are often realized via inner products, as L(f) = ⟨f, h⟩ for some fixed h.

Beyond functionals, mappings between function spaces—often called operators—play a central role. Linear operators T: X → Y satisfy T(f + g) = T(f) + T(g) and T(αf) = αT(f). In normed spaces, a linear operator is continuous if and only if it is bounded. Examples include multiplication operators (Mf)(x) = h(x)f(x), differentiation operators on suitable domains, and composition operators (C_φ f)(x) = f(φ(x)).

Key results relate to duality and representation. The Riesz representation theorem states that every continuous linear functional on a Hilbert space is given by an inner product with a unique element of the space. For L^p spaces (1 < p < ∞), the dual is L^q with 1/p + 1/q = 1; the dual of C_0 or continuous functions on a compact interval corresponds to measures.

Applications span calculus of variations, optimization, numerical analysis, and signal processing. The terms functionals and mappings are distinct yet closely related in functional analysis.