ON P-ABSOLUTELY SUMMING CONSTANTS OF BANACH SPACES

Given 1≦p<∞ and a real Banach space X, we define the p-absolutely summing constant μ p(X) as inf{Σ i =1/m |x*(x i)|p p Σ i =1/m {norm of matrix}x i{norm of matrix}p p]1 p}, where the supremum ranges over {x*∈X*; {norm of matrix}x*{norm of matrix}≤1} and the infimum is taken over all sets {x 1, x 2, ..., x m} ⊂X such that Σ i =1/m {norm of matrix}x i{norm of matrix}>0. It follows immediately from [2] that μ p(X)>0 if and only if X is finite dimensional. In this paper we find the exact values of μ p(X) for various spaces, and obtain some asymptotic estimates of μ p(X) for general finite dimensional Banach spaces. © 1969 Hebrew University.