The normed and banach envelopes of weakl1

Abstract

The space WeakL1 consists of all Lebesgue measurable functions on [0, 1] such that q(f) = supc>0 c λ{t : |f(t)| > c} is finite, where λ denotes Lebesgue measure. Let ρ be the gauge functional of the convex hull of the unit ball {f : q(f) ≤ 1} of the quasi-norm q, and let N be the null space of ρ. The normed envelope of WeakL1, which we denote by W, is the space (WeakL1/N, ρ). The Banach envelope of WeakL1, W̄, is the completion of W. We show that W̄ is isometrically lattice isomorphic to a sublattice of W. It is also shown that all rearrangement invariant Banach function spaces are isometrically lattice isomorphic to a sublattice of W.