nach oben

Erschienen in:

2019 | OriginalPaper | Buchkapitel

On Compacta K for Which C(K) Has Some Good Renorming Properties

In Honour of Manuel López-Pellicer

verfasst von : Aníbal Moltó

Erschienen in: Descriptive Topology and Functional Analysis II

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

By renorming it is usually meant obtaining equivalent norms in a Banach space with better properties, like being local uniformly rotund (LUR) or Kadets. In these notes we are concerned with C(K) spaces and pointwise lower semicontinuous Kadets or LUR renormings on them. If a C(K) space admits some of such equivalent norms then this space, endowed with the pointwise topology, has a countable cover by sets of small local norm-diameter (SLD). This property may be considered as the topological baseline for the existence of a pointwise lower semicontinuous Kadets, or even LUR renorming, since in many concrete cases it is the first step to obtain such a norm. In these notes we survey some methods, appearing in the literature, to prove that some C(K) spaces have this property.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Multiple Tensor Norms of Michor’s Type and Associated Operator Ideals

Nächstes Kapitel A Fixed Point Theory Linked to the Zeros of the Partial Sums of the Riemann Zeta Function

Alexandrov, G.: Locally uniformly convex norms in non separable Banach spaces, Ph.D. thesis, Kharkov (1980) (Russian)

Alexandrov, G.: Equivalent locally uniformly convex renorming of a Banach space of continuous functions on a compact topological group. Soviet Math. (Iz. VUZ) 26(1–7) (1982)

Argyros, S., Godefroy, G., Rosenthal, H.P.: Descriptive set theory and Banach spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1007–1069. Elsevier, Amsterdam (2003)CrossRef

Burke, M., Kubiś, W., Todorčević, S.: Kadec norms on spaces of continuous functions. Serdica Math. J. 32, 227–258 (2006)

Deville, R., Godefroy, G.: Some applications of projective resolutions of identity. Proc. London Math. Soc. 67, 183–199 (1993)MathSciNetCrossRef

Deville, R., Godefroy, G., Zizler, V.: Smoothness and Renormings in Banach Spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64. Longman Scientific & Technical, Longman House, Burnt Mill, Harlow (1993)

Engelking, R.: On functions defined in Cartesian products. Fundamenta Math. 59, 221–231 (1966)MathSciNetCrossRef

García, F., Oncina, L., Orihuela, J., Troyanski, S.: Kuratowski’s index of non-compactness and renorming in Banach spaces. J. Convex Anal. 11, 477–494 (2004)

Haydon, R.: A counterexample to several questions about scattered compact spaces. Bull. London Math. Soc. 22, 261–268 (1990)MathSciNetCrossRef

10.

Haydon, R.: Trees in renorming theory. Proc. London Math. Soc. 78, 549–584 (1999)MathSciNetCrossRef

11.

Haydon, R.: Locally uniformly rotund norms in Banach spaces and their duals. J. Funct. Anal. 254, 2023–2039 (2008)

12.

Haydon, R., Jayne, J., Namioka, I., Rogers, C.A.: Continuous functions on totally ordered spaces that are compact in their order topologies. J. Funct. Anal. 178, 23–63 (2000)MathSciNetCrossRef

13.

Haydon, R., Moltó, A., Orihuela, J.: Spaces of functions with countably many discontinuities. Israel J. Math. 158, 19–39 (2007)MathSciNetCrossRef

14.

Haydon, R., Rogers, C.A.: A locally uniformly convex renorming for certain \(C(K)\). Mathematika 37, 1–8 (1990)MathSciNetCrossRef

15.

Jayne, J., Namioka, I., Rogers, C.A.: \(\sigma \)-Fragmentable Banach spaces. Mathematika 39(161–188), 197–215 (1992)MathSciNetCrossRef

16.

Kechris, A.S.: Classical Descriptive Set Theory. Springer, New York (1995)

17.

Kubiś, W., Moltó, A., Troyanski, S.: Topological properties of the continuous function spaces on some ordered compacta. Set-Valued Var. Anal. 21, 649–659 (2013)MathSciNetCrossRef

18.

Martínez, J.F.: Renormings in \(C(K)\) spaces. Ph.D. thesis, Valencia (2007)

19.

Martínez, J.F.: Sigma-fragmentability and the property SLD in \(C(K)\) spaces. Topology Appl. 156, 1505–1509 (2009)

20.

Martínez, J.F., Moltó, A., Orihuela, J., Troyanski, S.: On locally uniformly rotund renormings in \(C(K)\) spaces. Can. J. Math. 62, 595–613 (2010)MathSciNetCrossRef

21.

Moltó, A., Orihuela, J., Troyanski, S., Valdivia, M.: A Non Linear Transfer Technique. Lecture Notes in Mathematics, vol. 1951. Springer, Berlin (2009)

22.

Raja, M.: Kadec norms and Borel sets in a Banach space. Studia Math. 136, 1–16 (1999)MathSciNetCrossRef

23.

Raja, M.: Locally uniformly rotund norms. Mathematika 46, 343–358 (1999)MathSciNetCrossRef

24.

Raja, M.: On dual locally uniformly rotund norms. Israel J. Math. 129, 77–91 (2002)MathSciNetCrossRef

25.

Smith, R.J, Troyanski, S.: Renormings of \(C(K)\) spaces. RACSAM 104, 375–412 (2010)MathSciNetCrossRef

26.

Todorcevič, S.: Topics in Topology. Lecture Notes in Mathematics, vol. 1652. Springer, New York (1997)

27.

Todorcevič, S.: Representing trees as relatively compact subsets of the first Baire class. Bull. Acad. Serbe Sci. Et Arts 131, 29–45 (2005)MathSciNetMATH

Titel: On Compacta K for Which C(K) Has Some Good Renorming Properties
verfasst von: Aníbal Moltó
Verlag: Springer International Publishing
Buch: Descriptive Topology and Functional Analysis II
Print ISBN: 978-3-030-17375-3

Electronic ISBN: 978-3-030-17376-0

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-030-17376-0_12

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"