In mathematics, Cartan's lemma refers to a number of results named after either Élie Cartan or his son Henri Cartan: In exterior algebra:* 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
In several complex variables: Let and and define rectangles in the complex plane C by
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.
In potential theory, a result that estimates the Hausdorff measure of the set on which a logarithmic Newtonian potential is small. See Cartan's lemma (potential theory).
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