---
tag: 1T23294NKC
kind: definition
name: Functional
msc: [49-XX, 46-XX]
defines: 1EQM0DJ4NC
uses: [000AYPA9AJ, 0EPAT51DXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Functional_(mathematics)
---

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $V$ be a [[vector space|000AYPA9AJ]] over $\mathbb{K}$.
A *functional* on $V$ is a [[function|0EPAT51DXQ]] $J \colon V \to \mathbb{K}$.
More generally, a functional on a subset $A \subseteq V$ is a function $J \colon A \to \mathbb{K}$.
In the calculus of variations, $V$ is usually a space of real functions, such as $C^1([a, b])$, and $J$ is real-valued.

## Remarks

Example: on $V = C([0, 1])$, the map $J(y) = \int_0^1 y(x)^2 \, dx$ is a real functional.
It is not linear, since $J(2y) = 4 J(y)$ while a linear map would give $2 J(y)$.
A functional that is linear is a [[linear form|07YQ1C78G0]]; when $V$ is normed, the continuous linear functionals form the [[dual space|07JBSZE0V4]] $V^*$.
Non-example: the map $y \mapsto y'$ on $C^1([0, 1])$ takes values in $C([0, 1])$, not in $\mathbb{K}$, so it is an operator and not a functional.
Some authors restrict the word to linear functionals; in the calculus of variations it means any scalar-valued map as above.