Complex extreme points and complex rotundity in Orlicz spaces equipped with the 𝑠-norm

Year 2023, Volume: 1 Issue: 1, 40 - 47, 21.06.2023

Esra Başar Badik Hüseyin Uysal , Şeyma Yaşar

https://doi.org/10.26650/ijmath.2023.00001

Abstract

Let Φ be an Orlicz function and 𝐿 Φ(𝑋, Σ, 𝜇) be the corresponding Orlicz space on a non-atomic, 𝜎-finite, complete measure space (𝑋, Σ, 𝜇). It is known that extreme points which are connected with rotundity of the whole spaces are the most essential and important geometric notion in the geometric theory of Banach spaces. On the other hand, geometric theory of complex Banach spaces has significant applications that differ from the geometric theory of real Banach spaces. In this paper, we first describe the complex extreme points of unit ball of Orlicz spaces equipped with the 𝑠-norm where 𝑠 is a strictly increasing outer function. We also give criteria for complex rotundity. Our study generalizes and unifies the results that have been obtained for the Orlicz norm and the 𝑝-Amemiya norm (1 < 𝑝 < ∞) separately.

Keywords

Orlicz space, complex extreme points, complex strictly rotund, s-norm

Thanks

We are grateful to anonymous referees for careful reading of the manuscript and for helpful comments.

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