Symmetrically separated sequences in the unit sphere of a Banach space
Abstract
We prove the symmetric version of Kottman's theorem, that is to say, we demonstrate that the unit sphere of an infinitedimensional Banach space contains an infinite subset $A$ with the property that $\x\pm y\ > 1$ for distinct elements $x,y\in A$, thereby answering a question of J. M. F. Castillo. In the case where $X$ contains an infinitedimensional separable dual space or an unconditional basic sequence, the set $A$ may be chosen in a way that $\x\pm y\ \geqslant 1+\varepsilon$ for some $\varepsilon > 0$ and distinct $x,y\in A$. Under additional structural properties of $X$, such as nontrivial cotype, we obtain quantitative estimates for the said $\varepsilon$. Certain renorming results are also presented.
 Publication:

arXiv eprints
 Pub Date:
 November 2017
 DOI:
 10.48550/arXiv.1711.05149
 arXiv:
 arXiv:1711.05149
 Bibcode:
 2017arXiv171105149H
 Keywords:

 Mathematics  Functional Analysis
 EPrint:
 19 pp