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

in ΛV. Then there are scalars h_ij = h_ji such that w i = ∑ j = 1 p h i j v j .

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

so that K 1 = K 1 ′ ∩ K 1 ″ . Let K₂, ..., K_n 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 so that again K = K ′ ∩ 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).