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:{{cite book ::\begin{align} K_1 &= { z_1=x_1+iy_1 | a_2 K_1' &= { z_1=x_1+iy_1 | a_1 K_1'' &= { z_1=x_1+iy_1 | a_2 \end{align} :so that K_1 = K_1'\cap K_1*. Let K2, ..., K**n'' be simply connected domains in C and let ::\begin{align} K &= K_1\times K_2\times\cdots \times K_n\ K' &= K_1'\times K_2\times\cdots \times K_n\ K &= K_1*\times K_2\times\cdots \times K_n \end{align} :so that again K = K'\cap K*. 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 F' in K' and F* in K'' 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).