We introduce an ordinal index which measures the complexity of a weakly null sequence, and show that a construction due to J. Schreier can be iterated to produce for each α < ω₁, a weakly null sequence in with complexity α. As in the Schreier example each of these is a sequence of indicator functions which is a suppression-1 unconditional basic sequence. These sequences are used to construct Tsirelson-like spaces of large index. We also show that this new ordinal index is related to the Lavrent’ev...
Many of the known complemented subspaces of have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well known complemented subspaces of . It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given by partitions...
A classical result of Cembranos and Freniche states that the C(K,X) space contains a complemented copy of c₀ whenever K is an infinite compact Hausdorff space and X is an infinite-dimensional Banach space. This paper takes this result as a starting point and begins a study of conditions under which the spaces C(α), α < ω₁, are quotients of or complemented in C(K,X).
In contrast to the c₀ result, we prove that if C(βℕ ×[1,ω],X) contains a complemented copy of then X contains a copy of c₀. Moreover,...
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