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, ..., vp are linearly independent elements of a vector space V and w1, ..., wp are such that
v 1 ∧ w 1 + ⋯ + v p ∧ w p = 0 {\displaystyle v_{1}\wedge w_{1}+\cdots +v_{p}\wedge w_{p}=0}
in ΛV. Then there are scalars hij = hji such that
w i = ∑ j = 1 p h i j v j . {\displaystyle 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
K 1 = { z 1 = x 1 + i y 1 | a 2 < x 1 < a 3 , b 1 < y 1 < b 2 } K 1 ′ = { 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 } {\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 K 1 = K 1 ′ ∩ K 1 ″ {\displaystyle K_{1}=K_{1}'\cap K_{1}''} . Let K2, ..., Kn be simply connected domains in C and let
K = K 1 × K 2 × ⋯ × K n K ′ = K 1 ′ × K 2 × ⋯ × K n K ″ = K 1 ″ × K 2 × ⋯ × K n {\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 K = K ′ ∩ K ″ {\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 F ′ {\displaystyle F'} in K ′ {\displaystyle K'} and F ″ {\displaystyle F''} in K ″ {\displaystyle K''} such that
F ( z ) = F ′ ( z ) F ″ ( z ) {\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).
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