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Characterizations of Rotundity and Smoothness by Approximate Orthogonalities

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Stypuła T., Wójcik P.

Annales Mathematicae Silesianae

2016

tom 30

nr 1

193-201

In this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity and smoothness of dual spaces.

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Geometric properties of Orlicz spaces equipped with \(p\)-Amemiya norms − results and open questions

100%

Wisła M.

Commentationes Mathematicae

2015

tom 55

nr 2

The classical Orlicz and Luxemburg norms generated by an Orlicz function \(\Phi\) can be defined with the use of the Amemiya formula [H. Hudzik and L. Maligranda, Amemiya norm equals Orlicz norm in general, Indag. Math. 11 (2000), no. 4, 573-585]. Moreover, in this article Hudzik and Maligranda suggested investigating a family of p-Amemiya norms defined by the formula \(\|x\|_{\Phi,p}=\inf_{k>0} \frac{1}{k} (1+I_\Phi^p(kx))^{1/p}\), where \(1\le p\le\infty\) (under the convention: \((1+u^\infty)^{1/\infty}=\lim_{p\to\infty}(1+u^p)^{1/p}=\max{1,u}\) for all \(u\ge 0\)). Based on this idea, a number of papers have been published in the past few years. In this paper, we present some major results concerning the geometric properties of Orlicz spaces equipped with p-Amemiya norms. In the last section, a more general case of Amemiya type norms is investigated. A few open questions concerning this theory will be stated as well.

Ograniczanie wyników

1 Annales Mathematicae Silesianae

1 Commentationes Mathematicae

1 Stypuła T.

1 Wisła M.

1 Wójcik P.

1 2016

1 2015

Wyniki wyszukiwania

Characterizations of Rotundity and Smoothness by Approximate Orthogonalities

Geometric properties of Orlicz spaces equipped with \(p\)-Amemiya norms − results and open questions