# 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:[1] Suppose that v1, ..., vp are linearly independent elements of a vector space V and w1, ..., wp are such that
${\displaystyle v_{1}\wedge w_{1}+\cdots +v_{p}\wedge w_{p}=0}$
in ΛV. Then there are scalars hij = hji such that
${\displaystyle w_{i}=\sum _{j=1}^{p}h_{ij}v_{j}.}$
{\displaystyle {\begin{aligned}K_{1}&=\{z_{1}=x_{1}+iy_{1}|a_{2}
so that ${\displaystyle K_{1}=K_{1}'\cap K_{1}''}$. Let K2, ..., Kn be simply connected domains in C and let
{\displaystyle {\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 ${\displaystyle K=K'\cap K''}$. 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 ${\displaystyle F'}$ in ${\displaystyle K'}$ and ${\displaystyle F''}$ in ${\displaystyle K''}$ such that
${\displaystyle F(z)=F'(z)F''(z)}$
in K.

## References

1. ^ *Sternberg, S. (1983). Lectures on Differential Geometry ((2nd ed.) ed.). New York: Chelsea Publishing Co. p. 18. ISBN 0-8218-1385-4. OCLC 43032711.
2. ^ Robert C. Gunning and Hugo Rossi (1965). Analytic Functions of Several Complex Variables. Prentice-Hall. p. 199.