Funktionale

Funktionale, in mathematics and related fields, refer to functionals — maps that assign a scalar value to a function. Typically, a functional F takes as input an element x from a function space X and outputs a scalar in the real or complex numbers. Functionals are central in functional analysis, where they are studied alongside spaces of functions.

Types and properties: A functional is linear if F(ax + by) = aF(x) + bF(y). When X is a normed space, a functional is bounded (equivalently continuous) if there exists a constant C with |F(x)| ≤ C ||x|| for all x in X. The set of all bounded linear functionals on X is called the dual space, denoted X*. For inner product spaces, the Riesz representation theorem identifies X* with the space itself via x ↦ ⟨x, ·⟩.

Examples: On L^p spaces, functionals of the form F(f) = ∫ f g, with fixed g in L^q, are linear and continuous for 1 ≤ p ≤ ∞. On spaces of continuous functions, evaluation at a point, F(f) = f(t0), is a linear functional.

Applications: Functionals are fundamental in optimization and the calculus of variations, where one minimizes or analyzes functionals S[φ]. In physics, action functionals assign a number to a field configuration. In economics and statistics, various objective functionals are studied.

Usage in language: In German mathematical literature, the term Funktional (plural Funktionale) is commonly used to denote such objects. Related concepts include dual spaces, the Hahn–Banach theorem, and weak topologies.