Near isometries of Bochner L¹ and L^∞ spaces

Michael James Cambern

Vol. 122 (1986), No. 1, 1–10

DOI: 10.2140/pjm.1986.122.1

Abstract

Let (Ω_i,Σ_i,μ_i) be σ-finite measure spaces, i = 1,2, and let E be a Hilbert space. If the Bochner spaces L^p(Ω₁,Σ₁,μ₁,E) and L^p(Ω₂,Σ₂,μ₂,E) are nearly isometric, for either p = 1 or p = ∞, then L¹(Ω₁,Σ₁,μ₁,E) is isometric to L¹(Ω₂,Σ₂,μ₂,E) and hence L^∞(Ω₁,Σ₁,μ₁,E) is isometric to L^∞(Ω₂,Σ₂,μ₂,E).

Mathematical Subject Classification 2000

Primary: 46B20

Secondary: 46E30, 46E40

Milestones

Received: 28 August 1984

Published: 1 March 1986