Cartan's lemma
# Cartan's lemma

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

       $v_{1}\wedge w_{1}+\cdots +v_{p}\wedge w_{p}=0$
    in ΛV. Then there are scalars $h_{ij}$ = $h_{ji}$ such that
       $w_{i}=\sum _{j=1}^{p}h_{ij}v_{j}.$
       ${\begin{aligned}K_{1}&=\{z_{1}=x_{1}+iy_{1}|a_{2}<x_{1}<a_{3},b_{1}<y_{1}<b_{2}\}\\K_{1}'&=\{z_{1}=x_{1}+iy_{1}|a_{1}<x_{1}<a_{3},b_{1}<y_{1}<b_{2}\}\\K_{1}''&=\{z_{1}=x_{1}+iy_{1}|a_{2}<x_{1}<a_{4},b_{1}<y_{1}<b_{2}\}\end{aligned}}$
    so that $K_{1}=K_{1}'\cap K_{1}''$. Let $K_2,$ ..., $K_n$ be simply connected domains in C and let
       ${\begin{aligned}K&=K_{1}\times K_{2}\times \cdots \times K_{n}\\K'&=K_{1}'\times K_{2}\times \cdots \times K_{n}\\K''&=K_{1}''\times K_{2}\times \cdots \times K_{n}\end{aligned}}$
    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.
