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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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