Counterexamples in rotundity of norms in Banach spaces

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F24%3A00378786" target="_blank" >RIV/68407700:21230/24:00378786 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1016/j.jmaa.2024.128455" target="_blank" >https://doi.org/10.1016/j.jmaa.2024.128455</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jmaa.2024.128455" target="_blank" >10.1016/j.jmaa.2024.128455</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Counterexamples in rotundity of norms in Banach spaces

Popis výsledku v původním jazyce

We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and Uniform Rotundity in Every Direction (URED). Our first three results show that we may distinguish between all of these three properties in every Banach space where such renormings are possible. Specifically, we show that in every infinite dimensional Banach space which admits a WUR (resp. URED) renorming, we can find a norm with the same condition and which moreover fails to be UR (resp. WUR). We prove that these norms can be constructed to be Locally Uniformly Rotund (LUR) in Banach spaces admitting such renormings. Additionally, we obtain that in every Banach space with a LUR norm we can find a LUR renorming which is not URED. These results solve three open problems posed by A.J. Guirao, V. Montesinos and V. Zizler. The norms we construct in this first part are dense. In the last part of this note, we solve a fourth question posed by the same three authors by constructing a C infinity -smooth norm in c 0 whose dual norm is not strictly convex. (c) 2024 Elsevier Inc. All rights reserved.

Název v anglickém jazyce

Counterexamples in rotundity of norms in Banach spaces

Popis výsledku anglicky

We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and Uniform Rotundity in Every Direction (URED). Our first three results show that we may distinguish between all of these three properties in every Banach space where such renormings are possible. Specifically, we show that in every infinite dimensional Banach space which admits a WUR (resp. URED) renorming, we can find a norm with the same condition and which moreover fails to be UR (resp. WUR). We prove that these norms can be constructed to be Locally Uniformly Rotund (LUR) in Banach spaces admitting such renormings. Additionally, we obtain that in every Banach space with a LUR norm we can find a LUR renorming which is not URED. These results solve three open problems posed by A.J. Guirao, V. Montesinos and V. Zizler. The norms we construct in this first part are dense. In the last part of this note, we solve a fourth question posed by the same three authors by constructing a C infinity -smooth norm in c 0 whose dual norm is not strictly convex. (c) 2024 Elsevier Inc. All rights reserved.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA23-04776S" target="_blank" >GA23-04776S: Interakce algebraických, metrických, geometrických a topologických struktur na Banachových prostorech</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Mathematical Analysis and Applications

ISSN

0022-247X

e-ISSN

1096-0813

Svazek periodika

538

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

16

Strana od-do

—

Kód UT WoS článku

001238441000001

EID výsledku v databázi Scopus

2-s2.0-85191478952