IJPAM: Volume 111, No. 1 (2016)

Title

RENORMINGS OF VASAK SPACES

Authors

Gaj Ram Damai$^1$, Prakash Muni Bajracharya$^2$, Yongjin Li$^3$
$^1$Siddhnath Science Campus (T.U.)
Mahendranagar, NEPAL
$^2$Central Department of Mathematics (T.U.)
Kirtipur, Kathmandu, NEPAL
$^3$Department of Mathematics
Sun Yat-Sen University
Guangzhou, 510275, P.R. CHINA

Abstract

In this paper, we present the basic facts about vasak (WCD) space with application to the construction of equivalent norm on the vasak spaces. We study nearly about an open problem raised in [1].

History

Received: September 1, 2016
Revised: October 9, 2016
Published: December 6, 2016

AMS Classification, Key Words

AMS Subject Classification: 46B22, 46B20
Key Words and Phrases: Asplund spaces, LUR norm, WCGs

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Bibliography

1
A. Amanollah, H. Haghshenas, Some classical and recent results concerning renorming theory. Thai J. Math., 10 (2012), no. 2, 481-495.

2
A. Aviles, Weakly countably determined spaces of high complexity. Studia Math., 185 (2008), no. 3, 291-303. https://dx.doi.org/10.4064/sm185-3-6

3
R. Deville, G. Godefroy, V. Zizler, Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics, 64. Longman Scientific and Technical, Harlow; copublished in the United States with John Wiley and Sons, Inc., New York, 1993.

4
J. Diestel, Geometry of Banach spaces��selected topics. Lecture Notes in Mathematics, Vol. 485. Springer-Verlag, Berlin-New York, https://dx.doi.org/10.1007/BFb0082079 1975.

5
M. Fabian, Each weakly countably determined Asplund space admits a Frechet differentiable norm. Bull. Austral. Math. Soc., 36 (1987), no. 3, 367-374. https://dx.doi.org/10.1017/S000497270000366X

6
M. Fabian, On a dual locally uniformly rotund norm on a dual Vasak space. Studia Math. 101 (1991), no. 1, 69-81.

7
M. Fabian, Gateaux differentiability of convex functions and topology. Weak Asplund spaces. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley and Sons, Inc., New York, 1997.

8
M. Fabian, P. Habala,P. H$\acute{a}$jek, V. S. Montesinos, J. Pelant, V. Zizler, Functional analysis and Infinite-dimensional geometry, Springer-Verlag, New York, 2001. https://dx.doi.org/10.1007/978-1-4757-3480-5

9
M. Fabian, G. Godefroy, V. Montesinos,and V. Zizle, Inner characterizations of weakly compactly generated Banach spaces and their relatives. J. Math. Anal. Appl., 297 (2004), no. 2, 419-455. https://dx.doi.org/10.1016/j.jmaa.2004.02.015

10
M. Fabian, V. Montesinos, V. Zizler, The Day norm and Gruenhage compacta. Bull. Austral. Math. Soc., 69 (2004), no. 3, 451-456. https://dx.doi.org/10.1017/S0004972700036236

11
M. Fabian, V. Montesinos and V. Zizler, Smoothness in Banach spaces. Selected problems. RACSAM. Rev. R. Acad. Cienc. Exactas F$\acute{i}$s. Nat. Ser. A Mat., 100 (2006), no. 1-2, 101-125.

12
M. Fabian, P. Habala, P. H$\acute{a}$jek, V. Montesinos, V. Zizler, Banach space theory. The basis for linear and nonlinear analysis. CMS Books in Mathematics/Ouvrages de Math$\acute{e}$matiques de la SMC. Springer, New York, 2011. https://dx.doi.org/10.1007/978-1-4419-7515-7

13
J. R. Giles, D. A. Gregory, and Brailey Sims, Characterisation of normed linear spaces with Mazur's intersection property. Bull. Austral. Math. Soc., 18 (1978), no. 1, 105-123. https://dx.doi.org/10.1017/S0004972700007863

14
P. Hajek, M. Johanis, Characterization of reflexivity by equivalent renorming. J. Funct. Anal., 211 (2004), no. 1, 163-172. https://dx.doi.org/10.1016/S0022-1236(03)00264-7

15
W. B. Johnson and J. Lindenstrauss. Handbook of the geometry of Banach spaces. Vol. 2. Edited by North-Holland, Amsterdam, 2003. unit 18, p 784-830.

16
A. Malto, J. Orihuela, S. Troyanski, M. Valdivia, A nonlinear transfer technique for renorming. Lecture Notes in Mathematics, 1951. Springer-Verlag, Berlin, 2009.

17
S. Mercourakis, Sophocles, On weakly countably determined Banach spaces. Trans. Amer. Math. Soc., 300 (1987), no. 1, 307-327.

18
L. Vasak, On one generalization of weakly compactly generated Banach spaces. Studia Math., 70 (1981), no. 1, 11-19.

19
D. Wilkins, The strong WCD property for Banach spaces. Internat. J. Math. Math. Sci., 18 (1995), no. 1, 67-70.

How to Cite?

DOI: 10.12732/ijpam.v111i1.4 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 111
Issue: 1
Pages: 31 - 42


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