functionales

Functionales, in mathematics, are mappings that assign a real or complex number to elements of a vector space. The term is often used for functionals, with a common emphasis on linear functionals: maps f: V → F that satisfy f(ax + by) = a f(x) + b f(y) for all vectors x, y in V and scalars a, b in F. When a functional is linear and continuous with respect to a given topology, it is typically called a continuous linear functional.

A central concept related to functionales is the dual space. For a vector space V over a field F, the dual space V* consists of all linear functionales on V. In finite-dimensional spaces, V* is naturally isomorphic to V; in infinite-dimensional spaces this duality becomes richer and depends on the topology imposed on V. In normed spaces, the continuous dual V' includes all linear functionals continuous under the norm.

Key results involving functionales include the Hahn-Banach theorem, which guarantees the extension of a linear functional from a subspace to the whole space without increasing its norm, and the Riesz representation theorem in Hilbert spaces, which states that every continuous linear functional is represented as an inner product with a unique vector.

Beyond linearity, the term functional is also used for maps that assign a scalar to a function or other object, as in calculus of variations where a functional maps a function to a real number (for example, an energy or action integral).

Functionales have broad applications across mathematics, including analysis, optimization, quantum mechanics, and economics, where they model measurements, objectives, and quantities dependent linearly on inputs.