“Cartan's lemma”的意思、由来-开放百科全书

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Cartan's lemma

释义

References

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, ..., 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

In several complex variables:^[2] Let {{nowrap|a₁ < a₂ < a₃ < a₄}} and {{nowrap|b₁ < b₂}} and define rectangles in the complex plane C by

so that . Let K₂, ..., K_n be simply connected domains in C and let

so that again . 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 in and in such that

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

1. ^*{{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}}
2. ^{{cite book | author = Robert C. Gunning and Hugo Rossi | title = Analytic Functions of Several Complex Variables | publisher = Prentice-Hall | year = 1965|pages= 199}}

1 : Lemmas

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