Functional (mathematics)

Common card · No. 290874

In functional analysis, a functional is a map from a vector space to its underlying field. Elements of this field are called scalars, typically real or complex numbers. If the map is linear, the functional is called a linear functional. In some cases, a linear form or linear functional is defined using a free module over a commutative ring. The definite integral is a linear functional on the space of continuous real-valued functions over the interval [a, b], as it linearly maps each function to a real number. In contrast, the norm of a vector on a normed space is a nonlinear functional. Beginning in the 17th century, mathematicians developed functionals to solve variational problems: finding the function representing a curve, surface, or path that minimizes or maximizes a specific measurement. This process requires assigning a single scalar to each possible function. Vito Volterra's functions of lines (fonctions de lignes) assigned a scalar to the function representing a curve...

Read on Wikipedia · Image 4C, CC BY-SA 3.0