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2 Bulletin of the Polish Academy of Sciences. Mathematics

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On some geometric properties of Banach spaces of continuous functions on separable compact lines

Michalak A.

Bulletin of the Polish Academy of Sciences. Mathematics

2017

Vol. 65, no. 1

57--68

We study properties of Banach spaces C(L) of all continuous scalar (real or complex) functions on compact lines L. First we show that if L is a separable compact line, then for every closed linear subspace X of C(L) with separable dual the quotient space C(L)/X possesses a sequence of continuous linear functionals separating its points. Next we show that for any compact line L the space C(L) contains no subspace isomorphic to a C(K) space where K is a separable nonmetrizable scattered compact Hausdorff space with countable height.

An isomorphic classification of C(2m x [0, α]) spaces

Galego E. M.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

Vol. 57, no 3-4

279-287

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