Hostname: page-component-f554764f5-246sw Total loading time: 0 Render date: 2025-04-11T12:30:22.699Z Has data issue: false hasContentIssue false
  • English
  • Français

A nonlinear map for midpoint locally uniformly rotund renorming

Published online by Cambridge University Press:  17 April 2009

S. Lajara  and
A.J. Pallars
S. Lajara
Affiliation:
Departamento de Matemticas, Universidad de Castilla La Mancha, Escuela Politcnica Superior de Albacete, Edificio Infante Don Juan Manuel, Campus Universitario, 02071 Albacete, Spain, email: [email protected]
A.J. Pallars
Affiliation:
Departamento de Matemticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide a criterion for midpoint locally uniformly rotund renormability of normed spaces involving the class of σ-slicely continuous maps, recently introduced by Moltó, Orihuela, Troyanski and Valdiva in 2003. As a consequence of this result, we obtain a theorem of G. Alexandrov concerning the three space problem for midpoint locally uniformly rotund renormings of Banach spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 1 , August 2005 , pp. 39 - 44
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Alexandrov, G., ‘On the three space problem for MLUR renorming of Banach spaces’, C. R. Acad. Bulgare Sci. 42 (1989), 17–20.Google Scholar
[2]Alexandrov, G. and Dimitrov, I., ‘On equivalent weakly midpoint locally uniformly rotund renormings of the space ℓ∞’, (in Russian), in Math. and Math. Education, Proc. 14th Spring Conference of the Union of Bulg. Mathematicians, Sunny Beach(Blgar. Akad. Nauk,Sofia,1985,), pp. 189–191.Google Scholar
[3]Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Appl. Math. 64 (Longman Scientific and Technical, Harlow, 1993).Google Scholar
[4]Godefroy, G., Troyanski, S., Whitfield, J. and Zizler, V., ‘Three space problem for locally uniformly rotund renormings of Banach spaces’, Proc. Amer. Math. Soc. 94 (1985), 647–652.CrossRefGoogle Scholar
[5]Haydon, R., ‘Trees in renorming theory’, Proc. London Math. Soc. 78 (1999), 541–585.CrossRefGoogle Scholar
[6]Hu, Z., Moors, W.B. and Smith, M.A., ‘On a Banach space without a weak mid-point locally uniformly rotund norm’, Bull. Austral. Math. Soc. 56 (1997), 193–196.CrossRefGoogle Scholar
[7]Sevilla, M. Jimnez and Moreno, J.P., ‘Renorming Banach spaces with the Mazur Intersection Property’, J. Funct. Anal. 144 (1997), 486–504.CrossRefGoogle Scholar
[8]Kunen, K. and Rosenthal, H., ‘Martingale proofs of some geometrical results in Banach space theory’, Pacific J. Math. 100 (1982), 153–175.CrossRefGoogle Scholar
[9]Molt, A., Orihuela, J. and Troyanski, S., ‘Locally uniformly rotund renorming and fragmentability’, Proc. London Math. Soc. 75 (1997), 619–640.CrossRefGoogle Scholar
[10]Molt, A., Orihuela, J., Troyanski, S. and Valdivia, M., ‘Midpoint locally uniformly rotundity and a decomposition method for renorming’, Q. J. Math. 52 (2001), 181–193.CrossRefGoogle Scholar
[11]Molt, A., Orihuela, J., Troyanski, S. and Valdivia, M., ‘A non linear transfer technique’, (Prepublicaciones del Departamento de Matemticas de la Universidad de Murcia no. 20, 2003).Google Scholar
[12]Raja, M., ‘On locally uniformly rotund norms’, Mathematika 46 (1999), 343–358.CrossRefGoogle Scholar