en>Cuzkatzimhut: /* First approach */

2014-12-05T16:46:30Z

First approach

en>ChrisGualtieri: /* First approach */Typo fixing, typos fixed: , → , using AWB

2012-08-09T04:12:56Z

First approach: Typo fixing, typos fixed: , → , using AWB

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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]]:<ref>*{{cite book | last = Sternberg | first = S. | year = 1983 | title = Lectures on Differential Geometry | edition = (2nd ed.) | publisher = Chelsea Publishing Co. | location = New York | isbn = 0-8218-1385-4 | oclc = 43032711 | page=18}}</ref> Suppose that ''v''1, ..., ''v''''p'' are linearly independent elements of a vector space ''V'' and ''w''1, ..., ''w''''p'' are such that
::<math>v_1\wedge w_1 + \cdots + v_p\wedge w_p = 0</math>
:in Λ''V''. 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]]:<ref>{{cite book
| author = [[Robert C. Gunning]] and [[Hugo Rossi]]
| title = Analytic Functions of Several Complex Variables
| publisher = Prentice-Hall
| year = 1965}}</ref> Let {{nowrap|''a''1 < ''a''2 < ''a''3 < ''a''4}} and {{nowrap|''b''1 < ''b''2}} 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 ''K''2, ..., ''K''''n'' 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 '''C'''''n'' 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)]].

==References==

{{reflist}}

[[Category:Lemmas]]

Rational difference equation - Revision history

en>Cuzkatzimhut: /* First approach */

en>ChrisGualtieri: /* First approach */Typo fixing, typos fixed: , → , using AWB