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Compact polynomials between Banach spaces. (English) Zbl 0853.46039

Summary: For every Banach space \(X\) we introduce indexes \(l(X)\) and \(u(X)\) in connection with the existence of upper and lower \(\ell_p\)-estimates for sequences in \(X\). These indexes are compared with type and cotype of space. We prove that polynomials preserve upper estimates of sequences, and we deduce that every polynomial \(P: X\to Y\) of degree \(N\) is compact when \(N\cdot u(Y)< l(X)\) and \(X\) does not contain a copy of \(\ell_1\).

MSC:

46G20 Infinite-dimensional holomorphy
46B20 Geometry and structure of normed linear spaces
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