is of bounded variation in if and only if it can be written as the difference ƒ = ƒ1 − ƒ2 of two non-decreasing functions on : this result is known as the Jordan decomposition of a function and it is related to the Jordan decomposition of a measure. Through the Stieltjes integral, any function of bounded variation on a closed interval [a, b] defines a bounded linear functional on C([a, b]). In this special case, the Riesz–Markov–Kakutani representation theorem states that every bounded linear functional arises uniquely in this way. The normalized positive functionals or probability measures correspond to positive non-decreasing lower semicontinuous functions. This point of view has been important in spectral theory, in particular in its application to ordinary differential equations. BV functions of several variables Functions of bounded variation, BV functions, are functions whose distributional derivative is a finite Radon measure. More precisely: Let be an open subset of . A function belonging to is said of bounded variation (BV function), and written if there exists a finite vector Radon measure such that the following equality holds that is, defines a linear functional on the space of continuously differentiable vector functions of compact support contained in : the vector measure represents therefore the distributional or weak gradient of . BV can be defined equivalently in the following way. Given a function belonging to , the total variation of in is defined as where is the essential supremum norm. Sometimes, especially in the theory of Caccioppoli sets, the following notation is used in order to emphasize that is the total variation of the distributional / weak gradient of . This notation reminds also that if is of class (i.e. a continuous and differentiable function having continuous derivatives) then its variation is exactly the integral of the absolute value of its gradient. The space of functions of bounded variation (BV functions) can then be defined as The two definitions are equivalent since if then therefore defines a continuous linear functional on the space . Since as a linear subspace, this continuous linear functional can be extended continuously and linearly to the whole by the Hahn–Banach theorem. Hence the continuous linear functional defines a Radon measure by the Riesz–Markov–Kakutani representation theorem. Locally BV functions If the function space of locally integrable functions, i.e. functions belonging to , is considered in the preceding definitions , and instead of the one