and studied in full generality by Władysław Orlicz and Julian Musielak in the paper : Laurence Chisholm Young studied earlier the case where is a positive integer. SBV functions SBV functions i.e. Special functions of Bounded Variation were introduced by Luigi Ambrosio and Ennio de Giorgi in the paper , dealing with free discontinuity variational problems: given an open subset of , the space is a proper linear subspace of , since the weak gradient of each function belonging to it consists precisely of the sum of an -dimensional support and an -dimensional support measure and no intermediate-dimensional terms, as seen in the following definition. Definition. Given a locally integrable function , then if and only if 1. There exist two Borel functions and of domain and codomain such that 2. For all of continuously differentiable vector functions of compact support contained in , i.e. for all the following formula is true: where is the -dimensional Hausdorff measure. Details on the properties of SBV functions can be found in works cited in the bibliography section: particularly the paper contains a useful bibliography. bv sequences As particular examples of Banach spaces, consider spaces of sequences of bounded variation, in addition to the spaces of functions of bounded variation. The total variation of a sequence x = (xi) of real or complex numbers is defined by The space of all sequences of finite total variation is denoted by bv. The norm on bv is given by With this norm, the space bv is a Banach space which is isomorphic to . The total variation itself defines a norm on a certain subspace of bv, denoted by bv0, consisting of sequences x = (xi) for which The norm on bv0 is denoted With respect to this norm bv0 becomes a Banach space as well, which is isomorphic and isometric to (although not in the natural way). Measures of bounded variation A signed (or complex) measure on a measurable space is said to be of bounded variation if its total variation is bounded: see , or the entry "Total variation" for further details. Examples As mentioned in the introduction, two large class of examples of BV functions are monotone functions, and absolutely continuous functions. For a negative example: the function is not of bounded variation on the interval While it is harder to see, the continuous function is not of bounded variation on the