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

${\displaystyle v_{1}\wedge w_{1}+\cdots +v_{p}\wedge w_{p}=0}$

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

${\displaystyle w_{i}=\sum _{j=1}^{p}h_{ij}v_{j}.}$

• In several complex variables:^[2] Let a₁ < a₂ < a₃ < a₄ and b₁ < b₂ and define rectangles in the complex plane C by

${\displaystyle {\begin{aligned}K_{1}&=\{z_{1}=x_{1}+iy_{1}|a_{2}<x_{1}<a_{3},b_{1}<y_{1}<b_{2}\}\\K_{1}'&=\{z_{1}=x_{1}+iy_{1}|a_{1}<x_{1}<a_{3},b_{1}<y_{1}<b_{2}\}\\K_{1}''&=\{z_{1}=x_{1}+iy_{1}|a_{2}<x_{1}<a_{4},b_{1}<y_{1}<b_{2}\}\end{aligned}}}$

so that ${\displaystyle K_{1}=K_{1}'\cap K_{1}''}$. Let K₂, ..., K_n 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.

• 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

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