Funktsionaal

Funktsionaal, or functional in English, is a map that assigns a single scalar value to a function. In mathematics, functionals typically act on spaces of functions and take values in the real or complex numbers. They are studied within functional analysis and appear in many areas, including calculus of variations, partial differential equations, and optimization. Functionals are defined on spaces such as C[a,b], L^p[a,b], or Sobolev spaces, and their properties depend on the topological structure of the domain.

A central distinction is between linear and nonlinear functionals. A functional F is linear if F(αf + βg) = αF(f) + βF(g) for all functions f, g in its domain and scalars α, β. If, in addition, there exists a constant C with |F(f)| ≤ C||f|| for all f, the functional is bounded (continuous). Examples of linear, bounded functionals include F(f) = ∫_a^b f(x) dx on L^p spaces and F(f) = ∫_a^b f(x)g(x) dx for a fixed function g. On C[a,b] with the sup norm, evaluation at a point, F(f) = f(x0), is another bounded linear functional with norm 1. On L^p spaces with 1 < p < ∞, point evaluation is not a continuous functional.

Representation and duality are key themes. In a Hilbert space H, every continuous linear functional F appears as F(f) = ⟨f, h⟩ for a unique h ∈ H (Riesz representation). More generally, the dual of L^p is L^q, where 1/p + 1/q = 1 (with special cases for p = 1 or ∞). Nonlinear functionals also occur, notably energy or action functionals in the calculus of variations, where the aim is to find a function that minimizes or extremizes the value of F. Functional analysis studies these maps, their continuity, and their implications for analysis and applications.