Vol. 52, No. 1, 1974

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Normed Köthe spaces as intermediate spaces of L1 and L

Stuart Edward Mills

Vol. 52 (1974), No. 1, 157–173

Let ,Σ) be a totally σ-finite measure space and let M(Δ) be the set of all complex-valued μ-measurable functions on Δ. This paper is concerned with determining whether certain classes of normed Köthe spaces (Banach function spaces) are intermediate spaces of L1 = L1(μ) and L = L(μ). It is proven that L1 L and L1 + L are associate Orlicz spaces and that for every nontrivial Young’s function Φ there is an equivalent Young’s function Φ1 such that the Orlicz space LuΦ1 is an intermediate space of L1 and L. The notion of a universal Köthe space is presented and it is proven that if Λ is a universal Köthe space then L1 LΛ L1 + L. Furthermore, if Λ is normed, in particular Λ = Lρ, then there is an equivalent universally rearrangement invariant norm ρ1 for which Lρ1 is an intermediate space of L1 and L.

Mathematical Subject Classification 2000
Primary: 46E30
Received: 18 January 1973
Revised: 28 November 1973
Published: 1 May 1974
Stuart Edward Mills