On weak compactness in L1 spaces

We will use the concept of strong generating and a simple renorming theorem to give new proofs to slight generalizations of some results of Argyros and Rosenthal on weakly compact sets in L1(μ) spaces for finite measures μ. The purpose of this note is to show that a simple transfer renorming theorem explains why L1(μ)-spaces, for finite measures μ, share some properties with superreflexive spaces, though there is no one-to-one bounded linear operator from L1(μ) into any reflexive space if L1(μ) is nonseparable [19, p. 232]. ∗Supported by grants IAA 100190610 and AVOZ 101 905 03 (Czech Republic). †Supported in part by Project MTM2005-08210 (Spain), the Generalitat Valenciana and the Universidad Politécnica de Valencia (Spain). ‡Supported by grants IAA 100 190 502 and AVOZ 101190503 (Czech Republic).