Abstract
This chapter is concerned with the following problem : how can we conveniently recognize when a map from (an open subset of) one vector space into another is invertible, and what regularity can we hope for in the inverse map? In fact it is very rare to be able to prove that the map is globally invertible, and we have to restrict ourselves to a ’local’ statement.
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Notes
This condition is automatically satisfied if the space under consideration is finite-dimensional. This is essentially the case which interests us; nevertheless the general case does not present any additional difficulty and is extremely useful.
The Norwegian mathematician Sophus LIE (1842-1899), who was professor at the University of Christiania (now Oslo) from 1872 to 1899, produced major work on infinite groups that was unrecognized during his lifetime. The Lie derivative is known by many names, notably particular derivative in mechanics and convective derivative in a slightly more general context.
Recall that if E and F are finite-dimensional then every linear map from E to F is continuous.
The expression “isomorphism of E onto F” means “bijection from E onto F with continuous inverse”; the latter property is a consequence of the former when E and F are finite-dimensional.
Hermann Amandus SCHWARZ (1843-1921) succeeded Weierstrass in 1892 at the University of Berlin.
It follows trivially from these definitions that a diffeomorphism is a local dif-feomorphism at every point. The converse is true in dimension 1 but false in dimension 2 as the example of the complex exponential shows.
Georg CANTOR (1845-1918), German mathematician, professor at Halle from 1879 to 1905, founder of the theory of infinite sets.
Richard D(UPEDEKIND) (1831-1916), German mathematician, professor at Brunswick Technical University from 1863 to 1894, was the inventor of (among other things) ideals and recursive functions.
Named after Jacob J(UPACOBI) (1804-1851), German mathematician, professorat Kœ-nigsberg from 1831 to 1848.
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© 2000 Springer-Verlag Berlin Heidelberg
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Demazure, M. (2000). Local Inversion. In: Bifurcations and Catastrophes. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57134-3_2
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DOI: https://doi.org/10.1007/978-3-642-57134-3_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-52118-1
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