A _functional_ is a [function](Analysis%20(index).md#Functions) that takes another function as an argument - and thus is a _function of a function_. Functionals are expressed with the square bracket notation, $F[f]$ in order to distinguish them from other functions. As functions of functions this means that the domain of a functional is a [function space.](Function%20spaces.md) Thus we define functionals as the following map, $F:\mathcal{\Gamma} \rightarrow \mathbb{F}, \;\;\;\; f\rightarrow F[f].$ That is, a functional is a map of an element, $\mathbf{f}$ of a function space, $\mathcal{\Gamma}$ onto a number field $\mathbb{F}$ where $\mathbb{F}$ is either $\mathbb{C}$ or $\mathbb{R}.$ %%Are functionals smooth maps? Note also that the "space" the function acts on is not necessarily a function space%% # Functionals of multiple functions In the same manner as [above](Functional.md) we may define functionals of set of functions, $F[f_1,f_2...f_n].$ %%Do these functions also belong in their respective function spaces?%% # Functionals as maps of trajectories In order to emphasize the notion of [functions as vectors](Functional%20analysis%20(index).md#Basic%20Concepts) we may rewrite the [Functional](Functional.md) as $F:\mathcal{\Gamma} \rightarrow \mathbb{F}, \;\;\;\; \mathbf{f}\rightarrow F[\mathbf{f}].$ where if $\mathbb{F}$ is the real numbers $\mathbb{R,}$ %%Here we need a modified version of the diagram on page 330 of von Delft's textbook%% %%Spin this sub-entry off into a new entry in the mechanics sections%% # Functionals as maps of a dual space to a number %%See Foland's real analysis pg 157%% # Real and Complex Functionals # local functional # non-local functional # variation of a functional In order to analyze the behavior of a functional in small deviation we look at its [variation](Variation%20of%20a%20functional.md), $\delta F[f_1,...f_n].$  # Functional derivative # Linear Functionals [[Linear functional]] --- # Proofs and Examples ## Examples of functionals ### smooth functions In the most trivial case a [Smooth function](Smooth%20function.md) $f(x)$ is itself a functional since it may be written as $F[f(y)]=\int_{-\infty}^{\infty} dx f(y) \delta(y-x) = f(x)$ where $\delta(y-x)$ is the [[Dirac Delta function]]. ### Definite integrals Definite [integral](Analysis%20(index)#Integral)s of [Continuous function](Continuous%20function.md)s are examples of functionals expressed as $F[f(x)]=\int_{a}^{b}dx\,f(x)$ And this is true for $n$ dimensional integrals as well. #### Length of a curve [[Length of a curve]] #### Area of a surface [[Area of a surface]] --- # Recommended Reading Since functionals are an important topic in many areas a physics, they are introduced in many mathematics textbooks aimed at physicists. A broad introduction to functionals may be found found in the following texts: * [Altland, A. von Delft, J. _Mathematics for Physicists_, Cambridge University Press, 2019](Altland,%20A.%20von%20Delft,%20J.%20Mathematics%20for%20Physicists,%20Cambridge%20University%20Press,%202019.md) pgs. 330-333. Here functionals are introduced in the context of the study of physical trajectories in space folllowed by an introduction to Lagrangian mechanics. Note here that functionals are introduced here as maps onto $\mathbb{R},$ but they may also map onto $\mathbb{C}.$ This an omission based on the desire to focus on the concept in a particular context - that context being, physical trajectories. * [Engel, E., Dreizler, R. M., _Density Functional Theory - An Advanced Course_, Springer (2011)]([Theoretical%20and%20Mathematical%20Physics]%20Eberhard%20Engel,%20Reiner%20M.%20Dreizler%20(auth.)%20-%20Density%20Functional%20Theory_%20An%20Advanced%20Course%20(2011,%20Springer-Verlag%20Berlin%20Heidelberg)%20-%20libgen.lc.pdf) pgs. 403-405. Here functionals are introduced as a map onto either a real or complex number that takes a function as an argument. This text follows this definition with an introduction to functional derivatives and variations. Here the author also references a another discussion by [Courant and Hilbert.](Functional.md#^e00652) This is an appendix to a larger text on density functional theory. This text is aimed at graduate students studying physics. * [Courant R., Hilbert D., _Methods of Mathematical Physics_ Vol. I, Wiley VCH 1989]([Wiley%20classics%20library]%20Courant,%20Richard_Hilbert,%20David%20-%20Methods%20of%20mathematical%20physics%20(1989,%20Wiley)%20-%20libgen.lc.pdf) pgs. 167-168 ^e00652 Functionals are also also introduced in a variety of texts covering linear algebra as well as functional analysis. #MathematicalFoundations/Analysis/Functions #MathematicalFoundations/Analysis/FunctionalAnalysis #MathematicalFoundations/Analysis/Functionals