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

$v_1\wedge w_1+\cdots +v_p\wedge w_p=0$

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

$w_i=\sum _{j=1}^p{h}_{ij}{v}_j.$

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

$\begin{array}{rl}{K}_1& =\{z_1=x_1+i{y}_1|a_2<x_1<a_3,b_1<y_1<b_2\}\\ {K_1}^{\prime }& =\{z_1=x_1+i{y}_1|a_1<x_1<a_3,b_1<y_1<b_2\}\\ {K_1}^{″}& =\{z_1=x_1+i{y}_1|a_2<x_1<a_4,b_1<y_1<b_2\}\end{array}$

so that $K_1={K_1}^{\prime }\cap {K_1}^{″}$. Let K₂, ..., K_n be simply connected domains in C and let

$\begin{array}{rl}K& =K_1\times K_2\times \cdots \times K_n\\ K^{\prime }& ={K_1}^{\prime }\times K_2\times \cdots \times K_n\\ K^{″}& ={K_1}^{″}\times K_2\times \cdots \times K_n\end{array}$

so that again $K=K^{\prime }\cap K^{″}$. 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 $F^{\prime }$ in $K^{\prime }$ and $F^{″}$ in $K^{″}$ such that

$F(z)=F^{\prime }(z)F^{″}(z)$

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).