Characterizations of elements of a double dual Banach space and their canonical reproductions
Tom 104 / 1993
Streszczenie
For every element x** in the double dual of a separable Banach space X there exists the sequence of the canonical reproductions of x** in the even-order duals of X. In this paper we prove that every such sequence defines a spreading model for X. Using this result we characterize the elements of X**╲ X which belong to the class B_1 (X)╲ B_{1/2}(X) (resp. to the class B_{1/4}(X)) as the elements with the sequence (x^{(2n)}) equivalent to the usual basis of ℓ^1 (resp. as the elements with the sequence (x^{(4n-2)} - x^{(4n)}) equivalent to the usual basis of c_0). Also, by analogous conditions but of isometric nature, we characterize the embeddability of ℓ^1 (resp. c_0) in X.