Abstract
We will now consider implicit function theorems in both the real analytic and the complex analytic (holomorphic) categories. These are obviously closely related, as the problem in the real analytic category can be complexified (by replacing every xj with a z j) and thereby turned into a holomorphic problem. Conversely, any complex analytic implicit function theorem situation is a fortiori real analytic and can therefore be treated with real analytic techniques. And both categories are subcategories of the C∞ category.
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Of course the algebraic operations must be required to be smooth or every Rn can be made a field trivially by using the one-to-one, onto mapping from set theory that demonstrates that R1 and Rn have the same cardinality.
A set B is balanced if \( cB \subseteq B \) holds for all scalars c with \( \left| c \right| \leqslant 1 \) 1. A set A is absorbent if \( Y = { \cup_{t > {0}}}tA \).
non-metrizable spaces. Our condition (2) also will hold for such complete spaces. The interested reader should consult Arkhangel’skiï and Fedorchuk [AF 90], Bourbaki [Bo 89], Page [Pa 78], or Zeidler [Ze 86]. The latter reference is specifically intended for the context of functional analysis.
The introduction of Nash [Na 56] discusses the history of the imbedding problem for Riemannian manifolds.
In the arguments that follow, we will apply this defining property of L for various values of k. Thus we are thinking of L as taking values in a space of pseudodifferential operators, which map Ck to \( {C^{k - m}} \) for every k.
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© 2003 Springer Science+Business Media New York
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Krantz, S.G., Parks, H.R. (2003). Advanced Implicit Function Theorems. In: The Implicit Function Theorem. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0059-8_6
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DOI: https://doi.org/10.1007/978-1-4612-0059-8_6
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6593-1
Online ISBN: 978-1-4612-0059-8
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