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Smooth Partitions of Unity on Banach Spaces
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approximation property arbitrary bounded sets bounded subsets chart compact completing the proof continuous function continuous on bounded continuously differentiable functions Corollary defined denote derivative differentiable manifolds dimensional Banach spaces directionally differentiable Edited exists f is uniformly finitely representable follows function f Further Geometry half-space Hausdorff space Hence Hilbert space homeomorphism homogeneous polynomial implies inequality infinite dimensional Banach integer isomorphic Let f locally finite manifolds modelled Math neighborhood nontrivial normed linear space open covering open set open subset partitions of unity polynomial of degree positive integer positive number preceding lemma preceding theorem Proceedings Proof Let real valued function reflexive reflexive space Remark S-categories S-partitions satisfying smooth Banach space smooth pair space F Sundaresan superreflexive spaces support of f supremum norm Theorem Let Theory topology U.C.D. function UF^-smooth ultrapower uniformly continuously differentiable uniformly convex uniformly F smooth unit ball verified Vlll weakly continuous