Cartan's lemma

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In mathematics, Cartan's lemma refers to a number of results named after either Élie Cartan or his son Henri Cartan:

  • In exterior algebra:[1] 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 ΛV. Then there are scalars hij = hji such that
w_i = \sum_{j=1}^p h_{ij}v_j.
\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, ..., Kn 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.

References[edit]

  1. ^ *Sternberg, S. (1983). Lectures on Differential Geometry ((2nd ed.) ed.). New York: Chelsea Publishing Co. p. 18. ISBN 0-8218-1385-4. OCLC 43032711. 
  2. ^ Robert C. Gunning and Hugo Rossi (1965). Analytic Functions of Several Complex Variables. Prentice-Hall.