An Isomorphic Classification of $C({\bf 2}^{\mathfrak{m}} \times [0, \alpha])$ Spaces

Elói Medina Galego¹

¹ Department of Mathematics University of São Paulo São Paulo, Brazil 05508-090

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 57 (2009) no. 3, pp. 279-287

Résumé

\def\mathfrak#1{{\mathfrak#1}}We present an extension of the classical isomorphic classification of the Banach spaces $C([0, \alpha])$ of all real continuous functions defined on the nondenumerable intervals of ordinals $[0, \alpha]$. As an application, we establish the isomorphic classification of the Banach spaces $C({\bf 2}^{\mathfrak{m}} \times [0, \alpha])$ of all real continuous functions defined on the compact spaces ${\bf 2}^{\mathfrak{m}} \times [0, \alpha]$, the topological product of the Cantor cubes ${\bf 2}^{\mathfrak{m}}$ with $\mathfrak{m}$ smaller than the first sequential cardinal, and intervals of ordinal numbers $[0, \alpha]$. Consequently, it is relatively consistent with ZFC that this yields a complete isomorphic classification of $C({\bf 2}^{\mathfrak{m}} \times [0, \alpha])$ spaces.

Détail

DOI : 10.4064/ba57-3-9

Keywords: def mathfrak mathfrak present extension classical isomorphic classification banach spaces alpha real continuous functions defined nondenumerable intervals ordinals alpha application establish isomorphic classification banach spaces mathfrak times alpha real continuous functions defined compact spaces mathfrak times alpha topological product cantor cubes mathfrak mathfrak smaller first sequential cardinal intervals ordinal numbers alpha consequently relatively consistent zfc yields complete isomorphic classification mathfrak times alpha spaces