Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T04:08:17.907Z Has data issue: false hasContentIssue false
  • English
  • Français

VECTOR SUBSPACES OF THE SET OF NON-NORM-ATTAINING FUNCTIONALS

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  01 June 2008

FRANCISCO J. GARCÍA-PACHECO
FRANCISCO J. GARCÍA-PACHECO*
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

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.

An example is found of a nonreflexive Banach space X such that the union of {0} and the set of non-norm-attaining functionals on X contains no two-dimensional subspace.

Keywords

lineablenon-norm-attaining functionalJames space

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces 46B03: Isomorphic theory (including renorming) of Banach spaces 46B07: Local theory of Banach spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 3 , June 2008 , pp. 425 - 432
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Acosta, M. D., Aizpuru, A., Aron, R. M. and García-Pacheco, F. J., ‘Functionals that do not attain their norm’, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), 407–418.CrossRefGoogle Scholar
[2]Aizpuru, A. and García-Pacheco, F. J., ‘A note on -summand vectors in dual spaces’, Glasgow Math. J. (2009), to appear.CrossRefGoogle Scholar
[3]Aron, R. M., Gurariy, V. I. and Seoane-Sepúlveda, J. B., ‘Lineability and spaceability of sets of functions on ℝ’, Proc. Amer. Math. Soc. 133 (2005), 795–803.CrossRefGoogle Scholar
[4]Aron, R. M., Pérez-García, D. and Seoane-Sepúlveda, J. B., ‘Algebrability of the set of non-convergent Fourier series’, Studia Math. 175 (2006), 83–90.CrossRefGoogle Scholar
[5]Bayart, F., ‘Topological and algebraic generecity of divergence and of universality’, Studia Math. 167 (2005), 161–181.CrossRefGoogle Scholar
[6]Bayart, F., ‘Linearity of sets of strange functions’, Michigan Math. J. 53 (2005), 291–303.CrossRefGoogle Scholar
[7]Bayart, F. and Quarta, L., ‘Algebras in sets of queer functions’, Israel J. Math 158 (2007), 285–296.CrossRefGoogle Scholar
[8]Bogachev, V. I., Mena-Jurado, J. F. and Navarro-Pascual, J. C., ‘Extreme points in spaces of continuous functions’, Proc. Amer. Math. Soc. 123 (1995), 1061–1067.CrossRefGoogle Scholar
[9]Deville, R., Godefroy, G. and Zizler, V., Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, 64 (John Wiley & Sons, New York, 1993).Google Scholar
[10]Diestel, J., Geometry of Banach Spaces–Selected Topics, Lecture Notes in Mathematics, 485 (Springer, Berlin, 1975).CrossRefGoogle Scholar
[11]García-Pacheco, F. J., Palmberg, N. and Seoane-Sepúlveda, J. B., ‘Lineability and algebrability of pathological phenomena in analysis’, J. Math. Anal. Appl. 326 (2007), 929–939.CrossRefGoogle Scholar
[12]Gurariy, V. I., ‘Linear spaces composed of everywhere non-differentiable functions’, C. R. Acad. Bulgare Sci. 44 (1991), 13–16.Google Scholar
[13]Gurariy, V. I. and Quarta, L., ‘On lineability of sets of continuous functions’, J. Math. Anal. Appl. 294 (2004), 62–72.CrossRefGoogle Scholar
[14]James, R. C., ‘Bases and reflexivity of Banach spaces’, Ann. Math. 52 (1950), 518–527.CrossRefGoogle Scholar
[15]James, R. C., ‘A non-reflexive Banach space which is isomorphic to its second conjugate’, Proc. Natl. Acad. Sci. USA 37 (1951), 174–177.CrossRefGoogle Scholar