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Uniformly convex space

About: Uniformly convex space is a research topic. Over the lifetime, 1842 publications have been published within this topic receiving 43552 citations.


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Journal ArticleDOI
01 Feb 1978
TL;DR: In this article, the authors give an answer to Ulam's problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist", and prove it for the case n = 1.
Abstract: Let E1, E2 be two Banach spaces, and let f: E1 -* E2 be a mapping, that is "approximately linear". S. M. Ulam posed the problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam's problem. THEOREM. Consider E1, E2 to be two Banach spaces, and let f: E1 -> E2 be a mapping such that f (tx) is continuous in t for each fixed x. Assume that there exists 0 > 0 andp E [0, 1) such that IIf(x + y) f (x) f(A)lI 0. The verification of (3) follows by induction on n. Indeed the case n = 1 is clear because by the hypothesis we can find 0, that is greater or equal to zero, andp such that 0 < p < 1 with 11[f(2x)]/2 -f(x)ll (4) IIxIIp Assume now that (3) holds and we want to prove it for the case (n + 1). However this is true because by (3) we obtain II [f (2n 2x)]/2 n f(2x)llI nI I*2x)2P < m E 2m(P therefore Received by the editors December 1, 1977. AMS (MOS) subject classifications (1970). Primary 47H15; Secondary 39A15.

2,694 citations

Book
01 Jan 1991

1,785 citations

Book
01 Jan 1999
TL;DR: The Radon-Nikodym property Negligible sets and Gateaux differentiability Lipschitz classification of Banach spaces Uniform embeddings into Hilbert space Uniform classification of spheres Uniform classifications of nonlinear quotient maps Oscillation of uniformly continuous functions on unit spheres of infinite-dimensional subspaces Perturbations of local and global isometries Twisted sums Group structure on Banach space as discussed by the authors.
Abstract: Introduction Retractions, extensions and selections Retractions, extensions and selections (special topics) Fixed points Differentiation of convex functions The Radon-Nikodym property Negligible sets and Gateaux differentiability Lipschitz classification of Banach spaces Uniform embeddings into Hilbert space Uniform classification of spheres Uniform classification of Banach spaces Nonlinear quotient maps Oscillation of uniformly continuous functions on unit spheres of finite-dimensional subspaces Oscillation of uniformly continuous functions on unit spheres of infinite-dimensional subspaces Perturbations of local isometries Perturbations of global isometries Twisted sums Group structure on Banach spaces Appendices Bibliography Index.

1,311 citations

Book
01 Jan 1975
TL;DR: Support functionals for closed bounded convex subsets of a Banach space are studied in this article, where the classical renorming theorems for the Radon-Nikodym theorem for vector measures are discussed.
Abstract: Support functionals for closed bounded convex subsets of a Banach space- Convexity and differentiability of norms- Uniformly convex and uniformly smooth Banach spaces- The classical renorming theorems- Weakly compactly generated banach spaces- The Radon-Nikodym theorem for vector measures

908 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202310
202225
20219
20201
20199
201810