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Let $\Lambda_{c}^{m}(\mathbb{R}^{n})$ denote the space of $C^{\infty}$ differentiable $m$-forms with compact support in $\mathbb{R}^{n}$. A continuous linear operator $T\colon\Lambda_{c}^{m}(\mathbb{R}^{n})\to\mathbb{R}$ is called an $m$-*current*. Let $\mathcal{D}_{m}$ denote the space of $m$-currents in $\mathbb{R}^{n}$.
We define a boundary operator $\partial\colon\mathcal{D}_{{m+1}}\to\mathcal{D}_{m}$ by

$\partial T(\omega):=T(d\omega).$ |

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:

$\int_{{\partial M}}\omega=\int_{M}d\omega.$ |

The space $\mathcal{D}_{m}$ of $m$-dimensional currents is a real vector space with operations defined by

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

The sum of two currents represents the *union* of the surfaces they represents. 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\ \text{whenever $\omega=0$ 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

$T_{k}(\omega)\to T(\omega),\qquad\forall\omega.$ |

A stronger norm on the space of currents is the *mass norm*. First of all we define the mass norm of a $m$-form $\omega$ as

$||\omega||:=\sup\{|\langle\omega,\xi\rangle|\colon\text{$\xi$ is a unit, % simple, $m$-vector}\}.$ |

So if $\omega$ is a simple $m$-form, then its mass norm is the usual norm of its coefficient. We hence define the *mass* of a current $T$ as

$\mathbf{M}(T):=\sup\{T(\omega)\colon\sup_{x}||\omega(x)||\leq 1\}.$ |

The mass of a currents represents the *area* of the generalized surface.

An intermediate norm, is the *flat norm* defined by

$\mathbf{F}(T):=\inf\{\mathbf{M}(A)+\mathbf{M}(B)\colon T=A+\partial B,\ A\in% \mathcal{E}_{m},\ B\in\mathcal{E}_{{m+1}}\}.$ |

Notice that two currents are close in the mass norm if they coincide apart from a small part. On the other hand the are close in the flat norm if they coincide up to a small deformation.

# Examples

Recall that $\Lambda_{c}^{0}(\mathbb{R}^{n})\equiv C^{\infty}_{c}(\mathbb{R}^{n})$ so that the following defines a $0$-current:

$T(f)=f(0).$ |

In particuar every signed measure $\mu$ with finite mass is a $0$-current:

$T(f)=\int f(x)\,d\mu(x).$ |

Let $(x,y,z)$ be the coordinates in $\mathbb{R}^{3}$. Then the following defines a $2$-current:

$T(a\,dx\wedge dy+b\,dy\wedge dz+c\,dx\wedge dz)=\int_{0}^{1}\int_{0}^{1}b(x,y,% 0)\,dx\,dy.$ |

## Mathematics Subject Classification

58A25*no label found*

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