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.

By Their Fruits Ye Shall Know Them: Some Remarks on the Interaction of General Topology with Other Areas of Mathematics

“…For that cardinal number the question is undecidable. Under MA + ¬CH, Pe lczyński's Conjecture is true for α = ω 1 as was also shown by Agryros [3]. But under CH, Haydon [82] constructed a counterexample of a particular nice form since it is of the form C(K) for a certain compact Hausdorff space K. The space K is an inverse limit of an ω 1 -sequence of Cantor sets with certain specific properties.…”

“…The answer to Pe lczyński's Conjecture is fascinating. For cardinals α > ω 1 it is true, as was shown by Agryros [3]. So there only remains the cardinal ω 1 .…”

“…Also here a survey of all significant results is impossible. There were in that period at least two major developments in general topology that revolutionized the field: the creations of infinite-dimensional topology and set theoretic topology 3 . It was mainly due to the efforts of Dick Anderson and Mary Ellen Rudin that these fields have played such a dominant role in general topology ever since.…”

On the existence of overcomplete sets in some classical nonseparable Banach spaces

“…First let us check that p ∈ P. Condition (a) -(c) of Definition 15 are clear. To prove condition (d) note that by (1) and by the choice of y i we have i<m |y i | ≤ δ q /2 + mθ, and so using ( 4) and ( 5) we conclude that (7) δ…”

“…(2) The spaces ℓ ∞ (λ) are isomorphic to the spaces C(βλ) respectively and βλ is extermally disconnected, so apply (1). The spaces L ∞ ({0, 1} λ ) are isomorphic to the spaces C(HY λ ) respectively, where HY λ is the Hewitt-Yosida space, i.e.…”