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  • Georges de Rham
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 266)

Abstract

In an n-dimensional manifold V, a current1 is a functional T [ø], defined on the vector space of all C forms ø with compact support in V, which is linear, that is, such that
$$T[{k_1}{\phi _1} + {k_2}{\phi _2}] = {k_1}T[{\phi _1}] + {k_2}T[{\phi _2}]$$
for all forms ø and ø of this space and for all constants k1and k2, and which is continuous in the following sense:

If ø h (h= 1, 2,…) is a sequence of C forms with supports all contained in a single compact set which is in the interior of the domain of a local coordinate system x1,…, x n such that each derivative of each coefficient of the form ø h (represented using x1,…, x n ) tends uniformly to zero as h→∞, then T [ø h ]→02.

Keywords

Vector Space Compact Support Topological Vector Space Bounded Subset Chain Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Georges de Rham
    • 1
  1. 1.LausanneSwitzerland

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