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In mathematics, the term functional is applied to certain functions. There are two common ways it is applied: these are related historically, but diverged somewhat during the twentieth century. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...
Partial plot of a function f. ...
(19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s The 20th century lasted from 1901 to 2000 in the Gregorian calendar (often from (1900 to 1999 in common usage). ...
Original meaning
The initial meaning is a function that takes functions as its argument; that is, a function whose domain is a set of functions. This was how the word was used initially, in the calculus of variations, where the integral to be minimized should be a functional, applied to an as-yet unknown function satisfying only some boundary conditions, and differentiability conditions. (See also operator, for a somewhat broader concept.) In mathematics, the domain of a function is the set of all input values to the function. ...
Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ...
In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ...
In mathematics, an operator is some kind of function; if it comes with a specified type of operand as function domain, it is no more than another way of talking of functions of a given type. ...
This usage still applies in that context and in many parts of physics and computer science, where in lambda calculus and functional programming a higher-order function is one that accepts a function and returns some value (or function). A black hole concept drawing by NASA. Physics (from the Greek, φυσικός (physikos), natural, and φÏσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ...
Computer science is the study of information and computation. ...
The lambda calculus is a formal system designed to investigate function definition, function application, and recursion. ...
The Haskell programming language logo Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions. ...
In mathematics and computer science, higher-order functions are functions which can take other functions as arguments, and may also return functions as results. ...
An example of a functional, in the physicist's sense, is
which turns any integrable function f into a real number, called the integral of f. Integrability is a mathematical concept used in different areas. ...
In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ...
The same usage also applies when one talks about a functional equation, meaning an equation between functionals: an equation F = G between functionals can be read as an 'equation to solve', with solutions being themselves functions. In such equations there may be several sets of variable unknowns, as when it is said that an additive function f is one satisfying the functional equation In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ...
- f(xy) = f(x) + f(y).
Linear functionals The secondary usage in the compound linear functional arises from functional analysis. While in the foundational period of functional analysis from 1900-1920, it was largely the study of vector spaces such as the Lp spaces that are function spaces, the later axiomatic approach made no such assumption. The name linear functional, however, was carried over and applied to the dual space construction, in the general case. In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ...
Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...
1900 (MCM) is a common year starting on Monday. ...
1920 (MCMXX) was a leap year starting on Thursday (link will take you to calendar) // Events January January 7 - Forces of Russian White admiral Kolchak surrender in Krasnoyarsk. ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ...
In mathematics, the existence of a dual vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). ...
Functional derivative and functional integration Functional derivatives are used in Lagrangian mechanics. They are derivatives of functionals: i.e. they carry information on how a functional (in the former sense, above) changes, when the function changes by a small amount. In mathematics and theoretical physics, the functional derivative is a generalization of the usual derivative that arises in the calculus of variations. ...
Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ...
Richard Feynman used functional integrals as the central idea in his sum over the histories formulation of quantum mechanics. This usage implies an integral taken over some function space. Richard Phillips Feynman (May 11, 1918 – February 15, 1988) (surname pronounced FINE-man; in IPA) was one of the most influential American physicists of the 20th century, expanding greatly the theory of quantum electrodynamics. ...
This article may be confusing for some readers, and should be edited to enhance clarity. ...
A simple introduction to this subject is provided in Basics of quantum mechanics. ...
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ...
See also: Distribution In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ...
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