Topology and Norms
The space of currents is naturally endowed with the weak-* topology, which will be further simply called weak convergence. A sequence Tk 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 Ļ is an m-form, then define its comass by
So if Ļ is a simple m-form, then its mass norm is the usual Lā-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) < ā 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.
Read more about this topic: Current (mathematics)
Famous quotes containing the word norms:
“For those parents from lower-class and minority communities ... [who] have had minimal experience in negotiating dominant, external institutions or have had negative and hostile contact with social service agencies, their initial approaches to the school are often overwhelming and difficult. Not only does the school feel like an alien environment with incomprehensible norms and structures, but the families often do not feel entitled to make demands or force disagreements.”
—Sara Lawrence Lightfoot (20th century)