Functional (mathematics)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a functional is a certain type of <a href="/facts/Function_(mathematics)/CdReEKCh">function</a>. The exact definition of the term varies depending on the subfield (and sometimes even the author).

<ul><li>In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, it is synonymous with a <a href="/facts/Linear_form/oGh7Asrz">linear form</a>, which is a linear mapping from a vector space 
 
 
 
 V
 
 
 {\displaystyle V}
 
 into its <a href="/facts/Field_(mathematics)/xAjAS4ko">field of scalars</a> (that is, it is an element of the <a href="/facts/Dual_space/4Uw4Knz1">dual space</a> 
 
 
 
 
 V
 
 ∗
 
 
 
 
 {\displaystyle V^{*}}
 
)</li>
<li>In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> and related fields, it refers to a mapping from a space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 into the field of <a href="/facts/Real_numbers/R02gw5Pb">real</a> or <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a>. In functional analysis, the term <a href="/facts/Linear_functional/oGh7Asrz">linear functional</a> is a synonym of <a href="/facts/Linear_form/oGh7Asrz">linear form</a>; that is, it is a scalar-valued linear map. Depending on the author, such mappings may or may not be assumed to be linear, or to be defined on the whole space 
 
 
 
 X
 .
 
 
 {\displaystyle X.}
 
</li>
<li>In <a href="/facts/Computer_science/lN3pEyPz">computer science</a>, it is synonymous with a <a href="/facts/Higher-order_function/VjnjGbpt">higher-order function</a>, which is a function that takes one or more functions as arguments or returns them.</li></ul>
This article is mainly concerned with the second concept, which arose in the early 18th century as part of the <a href="/facts/Calculus_of_variations/Ya5Lsma3">calculus of variations</a>. The first concept, which is more modern and abstract, is discussed in detail in a separate article, under the name <a href="/facts/Linear_form/oGh7Asrz">linear form</a>. The third concept is detailed in the computer science article on <a href="/facts/Higher-order_function/VjnjGbpt">higher-order functions</a>.
In the case where the space 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a space of functions, the functional is a "function of a function", and some older authors actually define the term "functional" to mean "function of a function".
However, the fact that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is a space of functions is not mathematically essential, so this older definition is no longer prevalent.
The term originates from the <a href="/facts/Calculus_of_variations/Ya5Lsma3">calculus of variations</a>, where one searches for a function that minimizes (or maximizes) a given functional. A particularly important application in <a href="/facts/Physics/a7aH2y08">physics</a> is search for a state of a system that minimizes (or maximizes) the <a href="/facts/Action_(physics)/NhWS2Mts">action</a>, or in other words the time integral of the <a href="/facts/Lagrangian_mechanics/sTxyqWz8">Lagrangian</a>.

Functional (mathematics) open-in-new

Functional (mathematics)