Transfinite Sequence Spaces
In this chapter, we discuss another biorthogonalization-like principle, namely Rosenthal’s principle of disjointization of measures. We then use this to prove the Pełczyński and Rosenthal results on nonweakly compact operators on C(K) spaces. We give Rosenthal’s characterization of C(K) spaces that contain a copy of \(c_0 \left( \Gamma \right)\) for uncountable \(\Gamma\). We then present results of Pełczyński, Talagrand, and others on characterizations of spaces containing a copy of \(\ell _1 \left( c \right)\). In the latter part of this chapter, we present characterizations of spaces with long unconditional bases that are weakly compactly generated (Johnson), weakly Lindelöf determined (Argyros, Mercourakis), or that admit uniformly Gâteaux differentiable norms (Troyanski). We also include some renorming results on spaces with long symmetric bases due to Troyanski.
KeywordsBanach Space Sequence Space Bounded Linear Operator Separable Banach Space Unconditional Basis
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