Journal of Convex Analysis 31 (2024), No. 1, 195--226
Copyright Heldermann Verlag 2024
Extensions of Continuous Linear Functionals and Smoothness of Banach Spaces
P. Gayathri
Dept. of Mathematics, National Institute of Technology, Tiruchirappalli, India
pgayathriraj1996@gmail.com
P. Shunmugaraj
Dept. of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India
psraj@iitk.ac.in
Vamsinadh Thota
Dept. of Mathematics, National Institute of Technology, Tiruchirappalli, India
vamsinadh@nitt.edu
We introduce a notion called strong extension property which is stronger than the well known Hahn-Banach extension property introduced by R. R. Phelps in 1960 and weaker than the uniform extension property introduced by V. Zizler in 1972. Characterizations of the strong extension property in terms of strong Chebyshevness and strong rotundity are presented in this article. We observe that the strong extension and uniform extension properties coincide when the extensions are considered on finite dimensional subspaces. We also obtain stability results pertaining to extension, strong extension and uniform extension properties. Further, for a subspace Y of a Banach space X, we consider different notions of smoothness of Y in X. These smoothness notions are characterized in terms of extension, strong extension and uniform extension properties.
Keywords: Extension property, strong extension property, strongly Chebyshev, smooth, Frechet smooth.
MSC: 46A22; 46B20, 41A65.
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