Functionals

Functionals are mappings that assign a single scalar value to each element of a space, typically a vector space over a field such as the real or complex numbers. Formally, if V is a vector space over F, a functional is a map f: V → F. When the functional is linear, it is called a linear functional, and the collection of all linear functionals on V forms its dual space V*.

A common example in finite-dimensional spaces is given by inner products: for a fixed vector a, the map v ↦ a · v is a linear functional. Functionals can also arise on spaces of functions. For a space F of functions, a functional may take a function as input and return a scalar, such as the evaluation functional ev_t: f ↦ f(t) or a more general integral functional J[f] = ∫ L(x, f(x), f′(x)) dx.

In calculus of variations, functionals assign numbers to functions and one seeks functions that extremize the value of the functional. The Euler–Lagrange equation provides necessary conditions for extrema. In functional analysis, attention is often given to continuity and boundedness: continuous linear functionals form the continuous dual, and the Riesz representation theorem identifies continuous linear functionals on a Hilbert space with inner products against a fixed element.

Functionals need not be linear or continuous; nonlinear functionals frequently arise in optimization and applications. In physics and economics, functionals include the action S[φ] in classical mechanics or the cost functional in control theory, each assigning a scalar quantity to a function or a trajectory.