Proceedings of the Centre for Mathematics and its Applications

Absolutely Chebyshev subspaces

Åsvald Lima and David Yost

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Abstract

Let's say that a closed subspace $M$ of a Banach space $X$ is absolutely Chebyshev if it is Chebyshev and, for each $x \in X, \| x \|$ can be expressed as a function of only $d(x,M)$ and $\| P_m(x)\|$. A typical example is a dosed subspace of a Hilbert space. Absolutely Chebyshev subspaces are, modulo renorming, the same as semi-$L$-summands. We show that any real Banach space can be absolutely Chebyshev in some larger space, with a nonlinear metric projection. Dually, it follows that if $M$ has the 2-ball property but not the 3-ball property in $X$, no restriction exists on the quotient space $X/M$. It is not known whether such examples can be found in complex Banach spaces.

Article information

Source
Workshop/Miniconference on Functional Analysis and Optimization. Simon Fitzpatrick and John Giles, eds. Proceedings of the Centre for Mathematical Analysis, v. 20. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1988), 116-127

Dates
First available in Project Euclid: 18 November 2014

Permanent link to this document
https://projecteuclid.org/ euclid.pcma/1416340108

Mathematical Reviews number (MathSciNet)
MR1009599

Zentralblatt MATH identifier
0673.41035

Citation

Lima, Åsvald; Yost, David. Absolutely Chebyshev subspaces. Workshop/Miniconference on Functional Analysis and Optimization, 116--127, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1988. https://projecteuclid.org/euclid.pcma/1416340108


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