Ja n 19 92 Isomorphism of certain weak L p spaces Denny

The Lorentz spaces play an important role in interpolation theory. They also form a class of Banach spaces generalizing the classical L spaces. In this paper, we continue the comparison of the Banach space structures of various weak L spaces begun in [2] and [3]. In [2], mimicking the construction of the Rademacher functions, it was shown that l can be embedded complementably into l. In [3], we showed that l can in turn be embedded complementably (even as a sublattice) into L[0, 1]. Here, we complete and extend these results by showing that, in fact, the three weak L spaces l, L[0, 1], and L[0,∞) are isomorphic as Banach spaces. This question was also mentioned in [1]. We start by recalling some standard definitions. Let (Ω,Σ, μ) be an arbitrary measure space. For 1 < p < ∞, the weak L space L(Ω,Σ, μ) is the space of all Σ-measurable functions f such that {ω : |f(ω)| > 0} is σ-finite and