functionale

Functionale is a term used in mathematics to denote a functional, a map that assigns a scalar to a function. In most contexts, a functional is a map F: V → F, where V is a vector space over a field F (typically the real or complex numbers). When F is linear, F is called a linear functional. If V is a normed or topological vector space and F is continuous (or bounded), it is called a bounded linear functional, and such functionals form the continuous dual space V*. In finite-dimensional spaces, every linear functional is continuous; in infinite-dimensional spaces, continuity is a substantive condition.

Examples include the evaluation functional at a point x0 on the space of functions on a domain D, defined by F(f) = f(x0); the integration functional F(f) = ∫_D f(x) w(x) dx with an integrable weight w; the derivative functional on differentiable functions, F(f) = f'(x0). In a Hilbert space, the Riesz representation theorem identifies continuous linear functionals with elements of the space via F(f) = ⟨f, g⟩.

Applications span functional analysis, optimization, and calculus of variations. One studies dual spaces, weak and weak-* topologies, and variational problems that minimize or maximize functionals, often producing Euler–Lagrange equations.

Terminology: In German-language literature, the term Funktional or Functionale is used in place of the English "functional." The concept extends to nonlinear functionals, which map functions to scalars but are not required to be linear.