on subspaces of the space of all currents. One such norm is the mass norm. If is an m-form, then define its comass by So if is a simple m-form, then its mass norm is the usual L∞-norm of its coefficient. The mass of a current is then defined as The mass of a current represents the weighted area of the generalized surface. A current such that M(T) < ∞ is representable by integration of a regular Borel measure by a version of the Riesz representation theorem. This is the starting point of homological integration. An intermediate norm is Whitney's flat norm, defined by Two currents are close in the mass norm if they coincide away from a small part. On the other hand, they are close in the flat norm if they coincide up to a small deformation. Examples Recall that so that the following defines a 0-current: In particular every signed regular measure is a 0-current: Let (x, y, z) be the coordinates in Then the following defines a 2-current (one of many): See also Georges de Rham Herbert Federer Differential geometry Varifold Notes References . Differential topology Functional analysis Generalized functions Generalized manifolds