Functional

Functional - a function defined on an arbitrary set and having a numerical range of values : usually a set of real numbers $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ $\ mathbb {R}$ or complex numbers $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ $\ mathbb {C}$ . In a broader sense, a functional is any mapping from an arbitrary set into an arbitrary (not necessarily numerical) ring .

Functionals are studied as one of the central concepts in functional analysis , and the main subject of calculus of variations is the study of variations of functionals.

Content

Definitions

The scope of a functional can be any set. If the domain is a topological space , we can define a continuous functional ; if the domain is a linear space over $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ or over $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ , one can define a linear functional ; if the domain of definition is an ordered set , a monotone functional can be defined.

Functionality defined on a topological space $X$ ${\ displaystyle X}$ is called continuous if it is continuous as a mapping into a topological space $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ or $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ .

Functionality defined on a topological space $X$ ${\ displaystyle X}$ is called continuous at $x\in X$ ${\ displaystyle x \ in X}$ if it is continuous at this point as a map to a topological space $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ or $\mathbb {C}$ ${\ displaystyle \ mathbb {C}}$ .

A functional defined on linear space and preserving addition and multiplication by a constant is called a linear functional . (The mapping of a linear space to a linear space is called an operator ).

One of the simplest functionalities is projection (mapping to a vector of one of its components or coordinates).

Quite often, the role of a linear space is played by a particular space of functions (continuous functions on a segment, integrable functions on a plane, etc.). Therefore, in applied areas, functional is often understood as a function of functions , a mapping that translates a function into a number (real or complex).

A functional on a linear space is called positive definite if its value is non-negative and equal to zero only at zero.

A mapping that converts a vector to its norm is a convex positive definite functional; this is one of the most common functionals. In physics, action is often used - also functional.

The optimization problems are formulated in the language of functionals : to find a solution (equations, systems of equations, systems of constraints, systems of inequalities, systems of inclusions and the like) that delivers an extremum (minimum or maximum) to a given functional. Functionals are also considered in the variational analysis .

Linear Functionality

Later, the concept of a functional in linear space was separated from the concept of a traditional functional, as a function that maps the elements of a linear space into its scalar space. Often (for example, when the space of functions is a linear space) these two varieties of the concept of “functional” coincide, at the same time they are not identical and do not absorb each other.

A particularly important variety of functionals is linear functionals .

Examples

function norm
fixed point function value
maximum or minimum function on the segment
value of the integral of the function
graph length of a real function of a real variable
the length of the curve parameterized by the vector function of the real argument (path length)
surface area parametrically specified by a vector function of two real arguments
scalar product by a fixed vector
action in mechanics
energy functional

Literature

V.I.Sobolev . Functional // Mathematical Encyclopedia / I.M. Vinogradov (Ch. Ed.). - M .: Soviet Encyclopedia, 1985. - V. 5. - 623 p. - 150,000 copies.
Kolmogorov A.N. , Fomin S.V. Elements of the theory of functions and functional analysis. - ed. fourth, recycled. - M .: Science , 1976 . - 544 p. - 35,000 copies.
U. Rudin. Functional analysis. - M .: Mir , 1975 .