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All Journal Indonesian Journal of Social and Environmental Issues (IJSEI) Indonesian Journal of Research and Educational Review (IJRER) Journal of Fundamental Mathematics and Applications (JFMA) Indonesian Journal Of Educational Research and Review Indonesian Journal of Mathematics and Applications
Wanjara, Amos Otieno
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A Classical Survey On Rotundity of Norms in Banach Spaces Wanjara, Amos Otieno
Indonesian Journal of Mathematics and Applications Vol. 2 No. 1 (2024): Indonesian Journal of Mathematics and Applications
Publisher : Universitas Brawijaya

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.21776/ub.ijma.2024.002.01.5

Abstract

The study on rotundity of norms in Banach spaces is of interest to many mathematicians. This is because rotundity properties are strongly applicable in many branches of Mathematics including but not restricted to functional analysis and topology. Some of the properties include: Rotund, Locally Uniformly Rotund, Uniformly Rotund in every direction, Uniformly Rotund in Weakly Compact sets of direction among others. In this paper, we give a survey of the standard results that will really revolutionize the study and research on rotundity of norms in the fields of functional analysis and topology among others.
ON A HIGHLY ROTUND NORM AND UNIFORMLY ROTUND NORM IN EVERY DIRECTION ON A FRECHE’T SPACE Wanjara, Amos Otieno
Journal of Fundamental Mathematics and Applications (JFMA) Vol 8, No 1 (2025)
Publisher : Diponegoro University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.14710/jfma.v8i1.24207

Abstract

The word rotund comes from Latin word "rotundus" implying wheel-shaped or round (from rota wheel). Rotundity is the roundness of a 3-dimensional object. Some of the properties of rotundity include: UR-Uniformly Rotund, LUR-Locally Uniformly Rotund, MLUR-Midpoint Locally Uniformly Rotund, WUR-Weakly Uniformly Rotund, URED-Uniformly Rotund in Every Direction, HR- Highly Rotund, WLUR-Weakly Locally Uniformly Rotund and URWC-Uniformly Rotund in Weakly Compact sets of directions. Problems on Rotundity properties are still open. Smith gave a summary chart on rotundity of norms in Banach spaces. The chart left an open question whether or not a Highly Rotund norm(HR) implies Uniformly Rotund norm on Every Direction(URED). It is not clear whether if a Banach space has a Highly Rotund(HR) norm it follows that it has and equivalently URED. In this paper, we investigated the relationship between a Highly Rotund norm(HR) and a Uniformly Rotund norm in Every Direction(URED) on a Freche’t Space. The result shows that both Highly Rotund norm and Uniformly Rotund norm on Every Direction(URED) exist in Freche’t spaces. The implication of this result is that rotundity properties can be extended within spaces. This research work is very important since rotundity properties are strongly applicable in many branches of mathematics.
Co-Authors Ogembo, Paul O Ogembo, Paul Okeyo