Abstract
The locally k-uniformly extremely convex Banach spaces are introduced. It is shown here that every locally k-uniformly extremely convex space is locally \((k+1)\)-uniformly extremely convex space, but the converse implication is not true. Some properties of locally k-uniformly extremely convex spaces are given. Specially, it is proved that if the second dual of Banach space X is locally 2-uniformly extremely convex, then X is reflexive. In addition, the notion of midpoint locally k-uniformly extreme convexity is introduced and its relations to other type convexity are given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.Bernal and F.Sullivan. Multi-dimensional volumes, super reflexivity and normal structure in Banach spaces. Illinois J. Math. 27(1983)501-513.
J.A.Clarkson. Uniformly convex spaces. Trans. Amer. Math. Soc. 40(1936)396-414.
M.M.Day. Normed linear spaces. (Springer-Verlag,Berlin-Heidelberg-New York, 1973).
K.Fan and I.Glicksburg. Fully convex normed linear spaces. Proc. Nat. Acad. (USA). 41(1955)683-698.
R.Geremia and F.Sullivan. Multi-dimensional volumes and moduli of convexity in Banach spaces. Annali de Math. Fure ed Applicata. 1(1981)1-21.
R.E.Huff. Banach spaces which are nearly uniformly convex. Rocky Mountain. J. Math. 10(1980)743-749.
D.N.Kutzarova. \(k\)-\(\beta\) and \(k\)-Nearly uniformly convex Banach spaces. J.Math.Anal.Appl. 162(1991) 322-338.
B.L.Lin and X.T.Yu. On the \(k\)-uniform rotund and the fully convex Banach spaces. J. Math. Anal. Appl. 110(1985)407-410.
A.R.Lovaglia. Locally uniformly convex Banach spaces. Trans. Amer. Math. Soc. 78(1955)225-238.
C.X.Nan and J.H.Wang. On the L\(k\)UR and L\(k\)R spaces. Math. Proc. Cambrige. Philos. Soc. 104(1988)521-526.
I.Singer. On the set of best approximation of an element in a normed linear space. Rev.Roumaine Math.Pures Appl. 5(1960) 383-402.
F.Sullivan. A generalization of uniformly rotund Banach spaces. Canad.J.Math. 31(1979)628-636.
S.Wulede and W.Ha. A new class of Banach space with the drop property. Proceedings of the Royal Society of Edinburgh. A142(2012)215-224.
S.Wulede, B.Wurichaihu and B.Wurina. A generalization of uniformly extremely convex Banach spaces. Journal of Function Space. 2016, Article ID 9161252.
X.T.Yu. On the locally \(k\)-uniform rotund spaces. Journal of East China Normal University. 4(1981)25-31, (Chinese).
Acknowledgements
This research was supported by the National Natural Science Foundation of China (Grant no. 11561053) . The author would like to thank the anonymous referee for some useful suggestions to improve the manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by T S S R K Rao.
Rights and permissions
About this article
Cite this article
Wulede, S. The locally k-uniformly extremely convex and midpoint locally k-uniformly extremely convex Banach spaces. Indian J Pure Appl Math 52, 512–520 (2021). https://doi.org/10.1007/s13226-021-00053-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-021-00053-4