Separated sets and Auerbach systems in Banach spaces

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00532935" target="_blank" >RIV/67985840:_____/20:00532935 - isvavai.cz</a>
Alternative codes found
RIV/68407700:21230/20:00346253
Result on the web
<a href="https://doi.org/10.1090/tran/8160" target="_blank" >https://doi.org/10.1090/tran/8160</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1090/tran/8160" target="_blank" >10.1090/tran/8160</a>

Result language
angličtina
Original language name
Separated sets and Auerbach systems in Banach spaces
Original language description
The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a large enough space $ X$, the unit sphere $ S_X$ always contains an uncountable $ (1+)$-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of weakly Lindelöf determined (WLD) spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of $ c_0(omega _1)$. Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically $ (1+varepsilon )$-separated subset of any regular cardinality not exceeding the density of $ X$, should the space $ X$ be super-reflexive, the unit sphere of $ X$ contains such a subset of cardinality equal to the density of $ X$. The said problem is studied for other classes of spaces too, including WLD spaces, RNP spaces, or strictly convex ones.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
Result was created during the realization of more than one project. More information in the Projects tab.
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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