mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Cartan's lemma refers to a number of results named after either

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He ...

or his son

Henri Cartan Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of c ...

: * In

exterior algebra In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...

: Suppose that ''v''₁, ..., ''v''_''p'' are linearly independent elements of a vector space ''V'' and ''w''₁, ..., ''w''_''p'' are such that ::

v_1\wedge w_1 + \cdots + v_p\wedge w_p = 0

:in Λ''V''. Then there are scalars ''h''_''ij'' = ''h''_''ji'' such that ::

w_i = \sum_^p h_v_j.

* In

several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties ...

: Let and and define rectangles in the complex plane C by ::

\begin
K_1 &= \ \\
K_1' &= \ \\
K_1'' &= \
\end

:so that

K_1 = K_1'\cap K_1''

. Let ''K''₂, ..., ''K''_''n'' be simply connected domains in C and let ::

\begin
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

:so that again

K = K'\cap K''

. 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

F'

K'

and

F''

K''

such that ::

F(z) = F'(z)F''(z)

:in ''K''. * In

potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that the two fundamental forces of nature known at the time, namely g ...

, a result that estimates the

Hausdorff measure In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assi ...

of the set on which a logarithmic Newtonian potential is small. See Cartan's lemma (potential theory).

References

Lemmas {{SIA, mathematics