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A Banach space C(K) picky about its biorthogonal systems.
- Speaker(s)
- Kamil Ryduchowski
- Affiliation
- University of Warsaw
- Language of the talk
- English
- Date
- Oct. 14, 2026, 4:15 p.m.
- Room
- room 4050
- Seminar
- Topology and Set Theory Seminar
For a non-metrizable zero-dimensional compact Hausdorff space K a biorthogonal system in C(K) is a sequence (f_alpha, mu_alpha) of pairs such that f_alpha is a continuous function on K, mu_alpha is a Radon measure on K and the integral of f_beta with respect to mu_alpha is equal either to 0 or to 1, with the latter happening if and only if alpha=beta.
Whereas it is consistent that every such C(K) admits an uncountable biorthogonal system with each mu_alpha being just the difference of two Dirac deltas, it is well known that the space of continuous functions on the Kunen's line admits no uncountable biorthogonal systems. We prove that, consistently, there is such a space K with C(K) admitting some uncountable biorthogonal systems, but not admitting such systems with all the measures atomic.
The idea of the construction is to start with the Cantor set and one-by-one split some omega_1 points into new Cantor sets.
