Cartan's lemma

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, ..., v_p are linearly independent elements of a vector space V and w₁, ..., w_p are such that

:

in ΛV. Then there are scalars h_ij = h_ji such that

:

• In several complex variables: Let and and define rectangles in the complex plane C by

:

so that . Let K₂, ..., K_n 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).

References