---
tag: 1EQM0DJ4NC
kind: concept
name: Functional
msc: [49-XX, 46-XX]
sources:
  - url: https://en.wikipedia.org/wiki/Functional_(mathematics)
same_as:
  wikidata: Q579600
  wikipedia: Functional (mathematics)
---

## Summary

A functional is a map that takes an element of a vector space, usually a function, and returns a number.
Length of a curve, energy of a field and the value of an integral are functionals of the curve, the field or the integrand.
Functionals are the objects that the [[calculus of variations|PQ25NHV0KS]] minimises and maximises.
The linear ones, the [[linear forms|07YQ1C78G0]], are the elements of the [[dual space|07JBSZE0V4]] and are central in functional analysis.

## Explanations

### L1 human

An ordinary function eats a number and gives a number.
A functional eats a whole function and gives a single number.
Think of a curve drawn between two points: its length is one number, so "length" is a functional on the set of curves.
For example, $J(y) = \int_0^1 y(x)^2 \, dx$ sends $y(x) = x$ to $1/3$ and $y(x) = 1$ to $1$.