is an m-dimensional current if it is
continuous in the following sense: If a sequence $\omega _{k}$ of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when $k$ tends to infinity, then $T(\omega _{k})$ tends to 0.
The space ${\mathcal {D}}_{m}(M)$ of m-dimensional currents on $M$ is a
realvector space with operations defined by
Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current $T\in {\mathcal {D}}_{m}(M)$ as the complement of the biggest
open set$U\subset M$ such that
$T(\omega )=0$
whenever $\omega \in \Omega _{c}^{m}(U)$
The
linear subspace of ${\mathcal {D}}_{m}(M)$ consisting of currents with support (in the sense above) that is a compact subset of $M$ is denoted ${\mathcal {E}}_{m}(M).$
Certain subclasses of currents which are closed under $\partial$ can be used instead of all currents to create a homology theory, which can satisfy the
Eilenberg–Steenrod axioms in certain cases. A classical example is the subclass of integral currents on Lipschitz neighborhood retracts.
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 $\omega$ is an m-form, then define its comass by
$\|\omega \|:=\sup\{\left|\langle \omega ,\xi \rangle \right|:\xi {\mbox{ is a unit, simple, }}m{\mbox{-vector}}\}.$
So if $\omega$ is a
simplem-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 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.