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

[math]\displaystyle{ v_1\wedge w_1 + \cdots + v_p\wedge w_p = 0 }[/math]

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

[math]\displaystyle{ w_i = \sum_{j=1}^p h_{ij}v_j. }[/math]

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

[math]\displaystyle{ \begin{align} K_1 &= \{ z_1=x_1+iy_1 | a_2 \lt x_1 \lt a_3, b_1 \lt y_1 \lt b_2\} \\ K_1' &= \{ z_1=x_1+iy_1 | a_1 \lt x_1 \lt a_3, b_1 \lt y_1 \lt b_2\} \\ K_1'' &= \{ z_1=x_1+iy_1 | a_2 \lt x_1 \lt a_4, b_1 \lt y_1 \lt b_2\} \end{align} }[/math]

so that [math]\displaystyle{ K_1 = K_1'\cap K_1'' }[/math]. Let K₂, ..., K_n be simply connected domains in C and let

[math]\displaystyle{ \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} }[/math]

so that again [math]\displaystyle{ K = K'\cap K'' }[/math]. 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 [math]\displaystyle{ F' }[/math] in [math]\displaystyle{ K' }[/math] and [math]\displaystyle{ F'' }[/math] in [math]\displaystyle{ K'' }[/math] such that

[math]\displaystyle{ F(z) = F'(z)F''(z) }[/math]

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).