Abstract
Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ p(X) as inf{Σ =1/mi |x*(x i)|p p Σ =1/mi ‖x i‖p p]1 p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x 1,x 2, …,x m} ⊂X such that Σ =1/mi ‖x i‖>0. It follows immediately from [2] thatμ p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ p(X) for various spaces, and obtain some asymptotic estimates ofμ p(X) for general finite dimensional Banach spaces.
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References
M. M. Day,Normed linear spaces, Springer-Verlag, Berlin, 1958.
A. Dvoretzky and C. A. Rogers,Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci.36 (1950), 192–197.
Y. Gordon,On the projection and Macphail constants of l pn spaces, Israel J. Math.6 (1968), 295–302.
B. Grünbaum,Projection constants, Trans. Amer. Math. Soc.95 (1960), 451–465.
F. John,Extremum problems with inequalities as subsidiary conditions, Courant Anniversary volume, 187–204, Interscience, New York (1948).
J. Lindenstrauss, and A. Pelczyńsky,Absolutely summing operators in ℒ p spaces and their applications, Studia Math.29 (1968), 275–326.
A. Pietsch,Absolut p-summierende Abbildungen in normierten Räumen, Studia Math.28 (1967), 333–353.
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This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.
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Gordon, Y. Onp-absolutely summing constants of banach spaces. Israel J. Math. 7, 151–163 (1969). https://doi.org/10.1007/BF02771662
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DOI: https://doi.org/10.1007/BF02771662