A note on smooth rotund norms which are not midpoint locally uniformly rotund

Carlo Alberto De Bernardi; A. Preti; J. Somaglia

doi:10.1016/j.jmaa.2025.129544

A note on smooth rotund norms which are not midpoint locally uniformly rotund

Carlo Alberto De Bernardi^*
, A. Preti
, J. Somaglia

^*Autore corrispondente per questo lavoro

Polytechnic University of Milan

Risultato della ricerca: Contributo in rivista › Articolo

Abstract

We prove that every separable infinite-dimensional Banach space admits a Gâteaux smooth and rotund norm which is not midpoint locally uniformly rotund. Moreover, by using a similar technique, we provide in every infinite-dimensional Banach space with separable dual a Fréchet smooth and weakly uniformly rotund norm which is not midpoint locally uniformly rotund. These two results provide a positive answer to some open problems by A. J. Guirao, V. Montesinos, and V. Zizler.

Lingua originale	Inglese
pagine (da-a)	N/A-N/A
Rivista	Journal of Mathematical Analysis and Applications
Volume	550
Numero di pubblicazione	2
DOI	https://doi.org/10.1016/j.jmaa.2025.129544
Stato di pubblicazione	Pubblicato - 2025

All Science Journal Classification (ASJC) codes

Analisi
Matematica Applicata

Keywords

Fréchet norm
Gâteaux norm
MLUR norm
Renorming
Rotund norm
URED norm

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10.1016/j.jmaa.2025.129544

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abstract = "We prove that every separable infinite-dimensional Banach space admits a G{\^a}teaux smooth and rotund norm which is not midpoint locally uniformly rotund. Moreover, by using a similar technique, we provide in every infinite-dimensional Banach space with separable dual a Fr{\'e}chet smooth and weakly uniformly rotund norm which is not midpoint locally uniformly rotund. These two results provide a positive answer to some open problems by A. J. Guirao, V. Montesinos, and V. Zizler.",

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T1 - A note on smooth rotund norms which are not midpoint locally uniformly rotund

AU - De Bernardi, Carlo Alberto

AU - Preti, A.

AU - Somaglia, J.

PY - 2025

Y1 - 2025

N2 - We prove that every separable infinite-dimensional Banach space admits a Gâteaux smooth and rotund norm which is not midpoint locally uniformly rotund. Moreover, by using a similar technique, we provide in every infinite-dimensional Banach space with separable dual a Fréchet smooth and weakly uniformly rotund norm which is not midpoint locally uniformly rotund. These two results provide a positive answer to some open problems by A. J. Guirao, V. Montesinos, and V. Zizler.

AB - We prove that every separable infinite-dimensional Banach space admits a Gâteaux smooth and rotund norm which is not midpoint locally uniformly rotund. Moreover, by using a similar technique, we provide in every infinite-dimensional Banach space with separable dual a Fréchet smooth and weakly uniformly rotund norm which is not midpoint locally uniformly rotund. These two results provide a positive answer to some open problems by A. J. Guirao, V. Montesinos, and V. Zizler.

KW - Fréchet norm

KW - Gâteaux norm

KW - MLUR norm

KW - Renorming

KW - Rotund norm

KW - URED norm

KW - Fréchet norm

KW - Gâteaux norm

KW - MLUR norm

KW - Renorming

KW - Rotund norm

KW - URED norm

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DO - 10.1016/j.jmaa.2025.129544

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SN - 0022-247X

VL - 550

SP - N/A-N/A

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -

A note on smooth rotund norms which are not midpoint locally uniformly rotund

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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