FUNCTIONALS Meaning and
Definition
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Functionals are mathematical objects that assign values to functions or set of functions. In other words, a functional is a rule or operation that takes in a function as input and produces a number or another function as output. The concept of functionals is widely used in various branches of mathematics, particularly in functional analysis and mathematical physics.
One common type of functional is the linear functional, which maps a function in a vector space to a scalar in the underlying field. Linear functionals can be seen as vectors in the dual space of the vector space. They are often used to study linear equations and linear operators.
Another important type of functional is the functional integral, used in the field of calculus of variations and quantum field theory. Functional integrals assign a number to a collection of functions, representing the sum or integral over all possible values of these functions. This concept has applications in statistical mechanics, quantum mechanics, and quantum field theory.
Functionals also find applications in optimization problems, where they are used to define and analyze objective functions. These functionals are often subject to constraints, and their minimization or maximization helps find optimal solutions to various real-world problems.
In summary, functionals are mathematical entities that associate values to functions or set of functions, playing a fundamental role in functional analysis, calculus of variations, quantum field theory, optimization problems, and other branches of mathematics and physics.
Etymology of FUNCTIONALS
The etymology of the word "functionals" can be traced back to the root word "function".
The word "function" originates from the Latin word "functio", meaning "performance" or "execution". It entered the English language in the 16th century.
The suffix "-al", derived from the Latin suffix "-alis", is used to form adjectives. It indicates being related, pertaining to, or characterized by something.
Therefore, "functionals" is the plural form of the adjective "functional", which is derived from "function". It refers to things or entities that have a particular purpose, utility, or perform a specific role.
