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Nonexpansive projections onto two-dimensional subspaces of Banach spaces. (English) Zbl 0634.46013
Let X be a Banach space and
(*) S a set of linearly independent smooth points in X such that every two-dimensional subspace intersecting S is the range of a nonexpansive projection.
If dim \(X=n\geq 3\) and \(card(S)=n-1\), then X is isometrically isomorphic to some \(\ell^ p(n)\) \((1<p\leq \infty)\). If dim \(X\geq 3\) and \(\overline{sp} S=X\), then either
(a) X is isometrically isomorphic to \(c_ 0(S)\) or some \(\ell^ p(S)\) with \(p\neq 2\) in such a way that each element of S corresponds to an element of the canonical basis or
(b) E is isometrically isomorphic to a Hilbert space.
A real Banach space X of dimension at least 3 is a Hilbert space iff there is S satisfying (*) and \(\| x+ty\| <\| x\|\) for some distinct x,y\(\in S\) and real t.
Reviewer: J.Danes
MSC:
46B25 Classical Banach spaces in the general theory
46B20 Geometry and structure of normed linear spaces
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References:
[1] Roberts, Convez Functions (1973)
[2] Lindenstrauss, Classical Banach Spaces II (1979) · doi:10.1007/978-3-662-35347-9
[3] DOI: 10.1007/BF01111244 · Zbl 0174.16801 · doi:10.1007/BF01111244
[4] Lacy, The Isometric Theory of Classical Banach Spaces (1974) · doi:10.1007/978-3-642-65762-7
[5] Blaschke, Leipziger Berichte 68 pp 50– (1916)
[6] Lindenstrauss, Classical Banach Spaces I (1977) · doi:10.1007/978-3-642-66557-8
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