Cartan's lemma

:in Λ*V*. Then there are scalars *h**ij* = *h**ji* such that ::w_i = \sum_{j=1}^p h_{ij}v_j. - 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 *K*2, ..., *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 **C***n* 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 ::F(z) = F'(z)F''(z) :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 ## References 1. Sternberg, S.. (1983). ["Lectures on Differential Geometry"](https://archive.org/details/lecturesondiffer00ster). *Chelsea Publishing Co.*. ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Cartan's_lemma) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Cartan's_lemma?action=history). ::