# Currents: The Deterministic Case

## Abstract

Currents are an extension of the concept of distributions, being themselves continuous linear functionals acting on a suitable space of differential forms; indeed, one might say that currents are differential forms having distributions as coefficients.

The first part of this chapter is devoted to a reminder of the main properties of distributions, as continuous linear functionals on the space of functions of compact support which are continuous with all possible derivatives. It is paid particular attention to the case of distributions associated with Radon measures.

Then the space of *m*-currents is introduced endowed with a suitable topology. Examples of distributions and currents are presented, with a particular attention to currents which anticipate the specific real applications presented in the relevant chapter.

Operations on currents and the definition of push-forward of a current are presented too.

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