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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Differentiable structures on function spaces

Author: Nishan Krikorian
Journal: Trans. Amer. Math. Soc. 171 (1972), 67-82
MSC: Primary 58D15; Secondary 58B10
MathSciNet review: 0312525
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Abstract: A $ {C^s}$ differentiable manifold structure is constructed for spaces of maps from a compact $ {C^r}$ manifold $ M$ to a $ {C^{r + s}}$ manifold $ N$. The method (1) is inspired by Douady; (2) does not require any additional structure on $ N$ (such as sprays); (3) includes the case when $ N$ is an analytic manifold and concludes that the mapping space is also an analytic manifold; (4) can be used to treat all the classical mapping spaces ($ {C^r}$ functions, $ {C^r}$ functions with Hölder conditions, and Sobolev functions). Several interesting aspects of these manifolds are investigated such as their tangent spaces, their behavior with respect to functions, and realizations of Lie group structures on them. Differentiable structures are also exhibited for spaces of compact maps with noncompact domain.

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Keywords: Differentiable structure, Banach manifold, exponential map, manifold model, differential map, analytic map, submersion, fibre product, transversal pair, Banach Lie group, foliation, Whitney extension, Hölder condition, Sobolev spaces, Calderón extension, compact map, Dugundji extension
Article copyright: © Copyright 1972 American Mathematical Society

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