On Nonseparable Banach Spaces

Abstract

Abstract. Combining combinatorial methods from set theory with the functional structure of certain Banach spaces we get some results on the isomorphic structure of nonseparable Banach spaces.

An uncountable version of Pták's combinatorial lemma

“…The complete solution to Pełczyński's conjecture follows from a combination of results due to Argyros and Haydon: Haydon [18] proved that the conjecture is false for κ = ω 1 and assuming CH. On the other hand, Argyros [2] proved the correctness of the conjecture for κ ω 2 (in ZFC) and for κ = ω 1 , assuming MA ω 1 . A different proof of Argyros' result can be obtained from [5]; let us also refer to [6,17,19,30] for a discussion of these and related results.…”