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2 Bulletin of the Polish Academy of Sciences. Mathematics

2 Kalenda O. F. K.

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Remark on the abstract Dirichlet problem for Baire-one functions

Kalenda O. F. K.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 1

55-73

We study the possibility of extending any bounded Baire-one function on the set of extreme points of a compact convex set to an affine Baire-one function and related questions. We give complete solutions to these questions within a class of Choquet simplices introduced by P. J. Stacey (1979). In particular we get an example of a Choquet simplex such that its set of extreme points is not Borel but any bounded Baire-one function on the set of extreme points can be extended to an affine Baire-one function. We also study the analogous questions for functions of higher Baire classes.

Note on Markushevich bases in subspaces and quotients of Banach spaces

Kalenda O. F. K.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 2

117-126

Let us call a Banach space Plichko (1-Plichko) if it admits a countably norming (countably 1-norming) Markushevich basis. We introduce a subclass of Plichko spaces strictly containing weakly Lindeloef determined spaces which is stable to isomorphisms, subspaces and quotients. Further we prove that each quotient of the space C(0, [omega]1) is 1-Plichko.