# 泛函

The arc length functional has as its domain the vector space of rectifiable curves (a subspace of ${\displaystyle C([0,1],\mathbb {R} ^{3})}$), and outputs a real scalar. This is an example of a non-linear functional.
The Riemann integral is a linear functional on the vector space of Riemann-integrable functions from ${\displaystyle \mathbb {R} }$ to ${\displaystyle \mathbb {R} }$.

${\displaystyle S\ }$是由一些函数構成的集合。所谓${\displaystyle S\ }$上的泛函就是${\displaystyle S\ }$上的一个实值函数。${\displaystyle S\ }$称为该泛函的容许函数集

## 性质

### 對偶性

${\displaystyle x_{0}\mapsto f(x_{0})}$

${\displaystyle f\mapsto f(x_{0})}$