Cartan's lemma explained

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

v _{1\wedge} w _{1} + … + v _{p\wedge} w _{p} = 0

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

w _{i} = \sum ^{p} _{j =1} h _{ij} v _{j.}

• In several complex variables:^[2] Let and and define rectangles in the complex plane C by

\begin{align} 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{align}}

so that

K _{1} = K _{1'\cap} K _{1''} . Let K₂, ..., K_n be simply connected domains in C and let\begin{align} K & = K _{1 x} K _{2 x …} x K _{n\\ K'} & = K _{1' x} K _{2 x …} x K _{n\\ K''} & = K _{1'' x} K _{2 x …} x K _{n \end{align}}

so that again

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' in K' and F'' in K'' such thatF(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).

Notes and References

1. • Book: Sternberg, S. . 1983 . Lectures on Differential Geometry . limited . (2nd ed.) . Chelsea Publishing Co. . New York . 0-8218-1385-4 . 43032711 . 18.
2. Book: . Analytic Functions of Several Complex Variables . Prentice-Hall . 1965. 199.