Cartan's lemma

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In mathematics, Cartan's lemma refers to a number of results named after either Élie Cartan or his son Henri Cartan:

  • In exterior algebra:[1] Suppose that v1, ..., vp are linearly independent elements of a vector space V and w1, ..., wp are such that
in ΛV. Then there are scalars hij = hji such that
so that . Let K2, ..., Kn be simply connected domains in C and let
so that again . Suppose that F(z) is a complex analytic matrix-valued function on a rectangle K in Cn such that F(z) is an invertible matrix for each z in K. Then there exist analytic functions in and in such that
in K.


  1. ^ *Sternberg, S. (1983). Lectures on Differential Geometry ((2nd ed.) ed.). New York: Chelsea Publishing Co. p. 18. ISBN 0-8218-1385-4. OCLC 43032711.
  2. ^ Robert C. Gunning and Hugo Rossi (1965). Analytic Functions of Several Complex Variables. Prentice-Hall. p. 199.