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Let \Lambda_c^m(\mathbb{R}^n) denote the space of Smooth ''m''-forms with Compact support on R''n''. A Continuous Linear Operator

:T\colon \Lambda_c^m(\mathbb{R}^n) o \mathbb{R}

is called an ''m''-current. Let \mathcal D_m denote the space of ''m''-currents in '''R'''''n''. We define a '''boundary''' Operator

:\partial\colon \mathcal D_{m+1} o \mathcal D_m

by

:\partial T(\omega) := T(d\omega).\,

giving a general form of Stokes' Theorem by definition. We will see that currents represent a generalization of ''m''-surfaces. In fact if ''M'' is a compact ''m''-dimensional oriented Manifold with boundary, we can associate to ''M'' the current ''M'' defined by

: M (\omega)=\int_M \omega.\,

So the definition of boundary \partial T of a current, is justified by Stokes Theorem on manifolds with boundary:

:\int_{\partial M} \omega = \int_M d\omega.\,

The space \mathcal D_m of ''m''-dimensional currents is a Real Vector Space with operations defined by

:(T+S)(\omega):= T(\omega)+S(\omega),\qquad (\lambda T)(\omega):=\lambda T(\omega).

Multiplication by a Scalar represents a change in the '' Multiplicity '' of the surface. In particular multiplication by −1 represents the change of Orientation of the surface.

We define the support of a current ''T'', denoted by

:\mathrm{spt}(T),\,

the smallest Closed Set ''C'' such that

:T(\omega) = 0\,

whenever ω = 0 on ''C''.

We denote with \mathcal E_m the Vector Subspace of \mathcal D_m of currents with compact support.


TOPOLOGY


The space of currents is naturally endowed with the ''weak-star'' Topology , which will be further simply called '' Weak Convergence ''. We say that a Sequence ''T''''k'' of currents, weakly Converges to a current ''T'' if

:T_k(\omega) o T(\omega),\qquad orall \omega.\,

A stronger Norm on the space of currents is the ''mass norm''. First of all we define the mass norm of a ''m''-form ω as