## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- 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

### Absolutely Chebyshev subspaces

#### 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

**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