Editing Functional (mathematics) (section)

==Details==

===Duality===
The mapping
<math display=block>x_0 \mapsto f(x_0)</math>
is a function, where <math>x_0</math> is an [[argument of a function]] <math>f.</math> 
At the same time, the mapping of a function to the value of the function at a point
<math display=block>f \mapsto f(x_0)</math>
is a ''functional''; here, <math>x_0</math> is a [[parameter]].

Provided that <math>f</math> is a linear function from a vector space to the underlying scalar field, the above linear maps are [[Duality (mathematics)|dual]] to each other, and in functional analysis both are called [[linear functional]]s.

===Definite integral===
[[Integral]]s such as
<math display="block">f\mapsto I[f] = \int_{\Omega} H(f(x),f'(x),\ldots) \; \mu(\mathrm{d}x)</math>
form a special class of functionals. They map a function <math>f</math> into a real number, provided that <math>H</math> is real-valued. Examples include
* the area underneath the graph of a positive function <math>f</math> <math display=block>f\mapsto\int_{x_0}^{x_1}f(x)\;\mathrm{d}x</math>
* [[Lp norm|<math>L^p</math> norm]] of a function on a set <math>E</math> <math display=block>f\mapsto \left(\int_E|f|^p \; \mathrm{d}x\right)^{1/p}</math>
* the [[arclength]] of a curve in 2-dimensional Euclidean space <math display=block>f \mapsto \int_{x_0}^{x_1} \sqrt{ 1+|f'(x)|^2 } \; \mathrm{d}x</math>

===Inner product spaces===
Given an [[inner product space]] <math>X,</math> and a fixed vector <math>\vec{x} \in X,</math> the map defined by <math>\vec{y} \mapsto \vec{x} \cdot \vec{y}</math> is a linear functional on <math>X.</math> The set of vectors <math>\vec{y}</math> such that <math>\vec{x}\cdot \vec{y}</math> is zero is a vector subspace of <math>X,</math> called the ''null space'' or ''[[Kernel (linear algebra)|kernel]]'' of the functional, or the [[orthogonal complement]] of <math>\vec{x},</math> denoted <math>\{\vec{x}\}^\perp.</math>

For example, taking the inner product with a fixed function <math>g \in L^2([-\pi,\pi])</math> defines a (linear) functional on the [[Hilbert space]] <math>L^2([-\pi,\pi])</math> of square integrable functions on <math>[-\pi,\pi]:</math>
<math display=block>f \mapsto \langle f,g \rangle = \int_{[-\pi,\pi]} \bar{f} g</math>

===Locality===
If a functional's value can be computed for small segments of the input curve and then summed to find the total value, the functional is called local. Otherwise it is called non-local. For example:
<math display=block>F(y) = \int_{x_0}^{x_1}y(x)\;\mathrm{d}x</math>
is local while
<math display=block>F(y) = \frac{\int_{x_0}^{x_1}y(x)\;\mathrm{d}x}{\int_{x_0}^{x_1} (1+ [y(x)]^2)\;\mathrm{d}x}</math>
is non-local. This occurs commonly when integrals occur separately in the numerator and denominator of an equation such as in calculations of center of mass.