Definition
Let denote the space of smooth m-forms with compact support on . A current is a linear functional on which is continuous in the sense of distributions. Thus a linear functional
is an m-current if it is continuous in the following sense: If a sequence of smooth forms, all supported in the same compact set, is such that all derivatives of all their coefficients tend uniformly to 0 when tends to infinity, then tends to 0.
The space of m-dimensional currents on ℝn is a real vector space with operations defined by
Multiplication by a constant scalar represents a change in the multiplicity of the surface. In particular multiplication by −1 represents the change of orientation of the surface.
Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current T as the complement of the biggest open set U such that T(ω) = 0 whenever the support of ω lies entirely in U.
The linear subspace of consisting of currents with compact support is denoted . It can be naturally identified with the dual space to the space of all smooth m-forms on ℝn.
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