point . V(·, Ω) is lower semi-continuous on L1(Ω) The functional is lower semi-continuous: to see this, choose a Cauchy sequence of BV-functions converging to . Then, since all the functions of the sequence and their limit function are integrable and by the definition of lower limit Now considering the supremum on the set of functions such that then the following inequality holds true which is exactly the definition of lower semicontinuity. BV(Ω) is a Banach space By definition is a subset of , while linearity follows from the linearity properties of the defining integral i.e. for all therefore for all , and for all , therefore for all , and all . The proved vector space properties imply that is a vector subspace of . Consider now the function defined as where is the usual norm: it is easy to prove that this is a norm on . To see that is complete respect to it, i.e. it is a Banach space, consider a Cauchy sequence in . By definition it is also a Cauchy sequence in and therefore has a limit in : since is bounded in for each , then by lower semicontinuity of the variation , therefore is a BV function. Finally, again by lower semicontinuity, choosing an arbitrary small positive number From this we deduce that is continuous because it's a norm. BV(Ω) is not separable To see this, it is sufficient to consider the following example belonging to the space : for each 0 < α < 1 define as the characteristic function of the left-closed interval . Then, choosing α,β∈ such that α≠β the following relation holds true: Now, in order to prove that every dense subset of cannot be countable, it is sufficient to see that for every it is possible to construct the balls Obviously those balls are pairwise disjoint, and also are an indexed family of sets whose index set is . This implies that this family has the cardinality of the continuum: now, since every dense subset of must have at least a point inside each member of this family, its cardinality is at least that of the continuum and therefore cannot a be countable subset. This example can be obviously extended to higher dimensions, and since it involves only local properties, it implies that the same property is true also for . Chain rule for BV functions Chain