Functional (mathematics)

Details

Duality

The mapping x_0 \mapsto f(x_0) is a function, where x_0 is an argument of a function f. At the same time, the mapping of a function to the value of the function at a point f \mapsto f(x_0) is a functional; here, x_0 is a parameter.

Provided that f is a linear function from a vector space to the underlying scalar field, the above linear maps are dual to each other, and in functional analysis both are called linear functionals.

Definite integral

Integrals such as f\mapsto I[f] = \int_{\Omega} H(f(x),f'(x),\ldots) ; \mu(\mathrm{d}x) form a special class of functionals. They map a function f into a real number, provided that H is real-valued. Examples include

• the area underneath the graph of a positive function f f\mapsto\int_{x_0}^{x_1}f(x);\mathrm{d}x
• L^p norm of a function on a set E f\mapsto \left(\int_E|f|^p ; \mathrm{d}x\right)^{1/p}
• the arclength of a curve in 2-dimensional Euclidean space f \mapsto \int_{x_0}^{x_1} \sqrt{ 1+|f'(x)|^2 } ; \mathrm{d}x

Inner product spaces

Given an inner product space X, and a fixed vector \vec{x} \in X, the map defined by \vec{y} \mapsto \vec{x} \cdot \vec{y} is a linear functional on X. The set of vectors \vec{y} such that \vec{x}\cdot \vec{y} is zero is a vector subspace of X, called the null space or kernel of the functional, or the orthogonal complement of \vec{x}, denoted {\vec{x}}^\perp.

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

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: F(y) = \int_{x_0}^{x_1}y(x);\mathrm{d}x is local while 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} 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.

Functional equations

Main article: Functional equation

The traditional usage also applies when one talks about a functional equation, meaning an equation between functionals: an equation F = G between functionals can be read as an 'equation to solve', with solutions being themselves functions. In such equations there may be several sets of variable unknowns, like when it is said that an additive map f is one satisfying Cauchy's functional equation: f(x + y) = f(x) + f(y) \qquad \text{ for all } x, y.

Derivative and integration

Functional derivatives are used in Lagrangian mechanics. They are derivatives of functionals; that is, they carry information on how a functional changes when the input function changes by a small amount.

Richard Feynman used functional integrals as the central idea in his sum over the histories formulation of quantum mechanics. This usage implies an integral taken over some function space.

References

References

1. {{harvnb. Lang. 2002
2. {{harvnb. Kolmogorov. Fomin. 1957
3. {{Harvard citation text. Axler. 2014 p. 101, §3.92
4. "Linear functional".
5. {{harvnb. Kolmogorov. Fomin. 1957