number, provided that is real-valued. Examples include the area underneath the graph of a positive function norm of a function on a set the arclength of a curve in 2-dimensional Euclidean space Inner product spaces Given an inner product space and a fixed vector the map defined by is a linear functional on The set of vectors such that is zero is a vector subspace of called the null space or kernel of the functional, or the orthogonal complement of denoted For example, taking the inner product with a fixed function defines a (linear) functional on the Hilbert space of square integrable functions on 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: is local while 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 The traditional usage also applies when one talks about a functional equation, meaning an equation between functionals: an equation 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 is one satisfying Cauchy's functional equation: 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. See also References Types of functions