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**1 - 1**of**1**### Spreading models isometric to l p or c 0 : The number of spreading models of a sequence

"... . We introduce methods to "count" the number of spreading models generated by a sequence and prove the following dichotomy. Every basic sequence in a Banach space X has a normalized block basis (xn ) which satisfies one and only one of the following two conditions: (1) (xn ) determines a s ..."

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. We introduce methods to "count" the number of spreading models generated by a sequence and prove the following dichotomy. Every basic sequence in a Banach space X has a normalized block basis (xn ) which satisfies one and only one of the following two conditions: (1) (xn ) determines a spreading model isometric to #p or c 0 over X; (2) There exists a separable Banach space Y that contains (xn ) and is finitely represented in X such that (xn ) admits continuum many subsequences which generate uniformly nonisomorphic spreading models over Y . 0. Introduction Let X be a separable Banach space. Every basic sequence (x n ) in X has a subsequence (y n ) such that the limit lim n0<<nk #x + # 0 y n0 + + # k y nk # exists for every choice of x # X and scalars # 0 , . . . , # k . The subsequence (y n ) determines a Banach space extension X of X called a spreading model of (x n ) over X. The space X is defined as follows. We let (a n ) be the standard unit vector basis of c 00 and...