A rank theorem for infinite dimensional spaces
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- by J. P. Holmes PDF
- Proc. Amer. Math. Soc. 50 (1975), 358-364 Request permission
Abstract:
Suppose $X$ is a Banach space, $U$ is an open set of $X$ containing 0, and $f$ is a continuously differentiable function from $U$ into $X$ satisfying $f(0) = 0$ and $f’{(0)^2} = f’(0)$. An additional hypothesis is given for $f$ which, in case $X$ is finite dimensional, is equivalent to assuming rank $f’(x) = \operatorname {rank} f’(0)$ for all $x$ in some neighborhood of 0. Under this hypothesis one obtains a local factorization of $f$ into ${h_1} \circ f’(0) \circ {h_2}$ where each of ${h_1}$ and ${h_2}$ is a continuously differentiable homeomorphism. In addition there is a neighborhood of 0 in ${f^{ - 1}}(\{ 0\} )$ which is the image of a continuously differentiable retraction. An application of these results to the theory of differentiable multiplications is given.References
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- J. P. Holmes, Differentiable projections and differentiable semigroups, Proc. Amer. Math. Soc. 41 (1973), 251–254. MR 375378, DOI 10.1090/S0002-9939-1973-0375378-1
- J. P. Holmes, Differentiable semigroups, Colloq. Math. 32 (1974), 99–104. MR 369599, DOI 10.4064/cm-32-1-99-104
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Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 50 (1975), 358-364
- MSC: Primary 58C25
- DOI: https://doi.org/10.1090/S0002-9939-1975-0383452-0
- MathSciNet review: 0383452