Polish spaces of Banach spaces: Complexity of isometry and isomorphism classes

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F24%3A00599029" target="_blank" >RIV/67985840:_____/24:00599029 - isvavai.cz</a>
Alternative codes found
RIV/00216208:11320/24:10492949
Result on the web
<a href="https://doi.org/10.1017/S1474748023000440" target="_blank" >https://doi.org/10.1017/S1474748023000440</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1017/S1474748023000440" target="_blank" >10.1017/S1474748023000440</a>

Result language
angličtina
Original language name
Polish spaces of Banach spaces: Complexity of isometry and isomorphism classes
Original language description
We study the complexities of isometry and isomorphism classes of separable Banach spaces in the Polish spaces of Banach spaces, recently introduced and investigated by the authors in [14]. We obtain sharp results concerning the most classical separable Banach spaces. We prove that the infinite-dimensional separable Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinitedimensional Banach space whose isomorphism class is Fσ. For p ∈ [1,2) (2,∞), we show that the isometry classes of Lp[0,1] and ℓp are G-complete sets and Fσ-complete sets, respectively. Then we show that the isometry class of c0 is an Fσ-complete set. Additionally, we compute the complexities of many other natural classes of separable Banach spaces, for instance, the class of separable Lp,λ+-spaces, for p,λ ≥ 1, is shown to be a G-set, the class of superreflexive spaces is shown to be an Fσ-set, and the class of spaces with local Π-basis structure is shown to be a Σ06 -set. The paper is concluded with many open problems and suggestions for a future research.
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
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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