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, ..., vp are linearly independent elements of a vector space V and w1, ..., wp are such that
 * $$v_1\wedge w_1 + \cdots + v_p\wedge w_p = 0$$
 * in &Lambda;V. Then there are scalars hij = hji 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{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\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{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''$$ such that
 * $$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).