Current (mathematics) - Topology and Norms

Topology and Norms

The space of currents is naturally endowed with the weak-* topology, which will be further simply called weak convergence. A sequence T_k of currents, converges to a current T if

It is possible to define several norms on subspaces of the space of all currents. One such norm is the mass norm. If ω is an m-form, then define its comass by

So if ω is a simple m-form, then its mass norm is the usual L∞-norm of its coefficient. The mass of a current T 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 over a suitably weighted rectifiable submanifold. 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.