The approximation and extension of uniformly continuous Banach space valued mappings
Year: 1983
Type: other
Abstract: i The aim of this paper is to present several results concerning the extension of uniformly continuous mappings, and the approximation of uniformly continuous mappings by Holder mappings in the setting of infinite-dimensional Banach spaces.In particular, we show that every uniformly continuous mapping between Hilbert spaces can be uniformly approximated by Lipschitz mapping* Key words and Phrasest Uniformly continuous, equi-uniformly continuous, Lipschitz, Holder, Lipschitz for large distances, modulus of continuity, c (I), ^(D» L p .Classification!Primary 26A16, 41A30, 54C20 Secondary 54E15 The principal results of the paper are the following.Theorem 1 establishes the equivalence of extending a uniformly continuous mapping with values in an ell-infinity space and uniformly approximating the mapping by Lipschitz mappings, while Theorem 3 establishes a similar result for uniformly continuous mappings between subsets of Hilbert spaces.Corollary 2 establishes that every uniformly continuous mapping between Hilbert spaces can be uniformly approximated by Lipsohitz mappings.Finally, Corollary 3 characterizes the subsets of Hilbert space for which every uniformly continuous Hilbert space valued mapping can be extended; these are precisely the U-embedded sets studied in [LR^ -.This result
Authors: Ronnie Levy, Michael Rice
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2598087095