Английская Википедия:Cartan's lemma

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

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>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>w_i = \sum_{j=1}^p h_{ij}v_j.</math>

• In several complex variables: Let Шаблон:Nowrap and Шаблон:Nowrap and define rectangles in the complex plane C by

<math>\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}</math>

so that <math>K_1 = K_1'\cap K_1</math>. Let K₂, ..., Kn be simply connected domains in C and let

<math>\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>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>F'</math> in <math>K'</math> and <math>F</math> in <math>K</math> such that

<math>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).