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 ''v''₁, ..., ''v''_''p'' are linearly independent elements of a vector space ''V'' and ''w''₁, ..., ''w''_''p'' are such that
::$v_1\wedge w_1 + \cdots + v_p\wedge w_p = 0$
:in Λ''V''. Then there are scalars ''h''_''ij'' = ''h''_''ji'' such that
::$w_i = \sum_^p h_v_j.$

* In several complex variables: Let and and define rectangles in the complex plane C by
::$\begin K_1 &= \ \\ K_1' &= \ \\ K_1'' &= \ \end$
:so that $K_1 = K_1'\cap K_1''$. Let ''K''₂, ..., ''K''_''n'' be simply connected domains in C and let
::$\begin K &= K_1\times K_2\times\cdots \times K_n\\ K' &= K_1'\times K_2\times\cdots \times K_n\\ K'' &= K_1''\times K_2\times\cdots \times K_n \end$
:so that again $K = K'\cap K''$. Suppose that ''F''(''z'') is a complex analytic matrix-valued function on a rectangle ''K'' in C^''n'' 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 that
::$F(z) = F'(z)F''(z)$
: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

Lemmas
{{SIA, mathematics