M-bases in spaces of continuous functions on ordinals. (English) Zbl 1029.46006

Kalenda, Ondřej F. K.

Colloq. Math. 92, No. 2, 179-187 (2002).

An uncountable M-basis \((x_i, f_i)_{i \in I}\) of a Banach space \(X\) is said to be countably norming, if the subspace \[ S=\{f \in X^*: \{i \in I: f(x_i) \neq 0 \} \;\text{ is} \;\text{ countable}\} \] is norming. The main result of the present paper is that the space \(C[0, \omega_2]\) has no countably norming M-basis. This answers a question by G. Alexandrov and A. Plichko.

Reviewer: Vladimir Kadets (Kharkov)

MSC:

46B04	Isometric theory of Banach spaces
46B26	Nonseparable Banach spaces
46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Keywords:

Markushevich bases; spaces of continuous functions on ordinals