Vol. 66, No. 1, 1976

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A representation of additive functionals on Lp-spaces, 0 < p < 1

Judith Ann Palagallo

Vol. 66 (1976), No. 1, 221–234
Abstract

Mizel and Sundaresan have given an integral representation for a class of nonlinear functionals, called additive functionals, on the Banach spaces Lp, p 1. In this paper, analogous results for these additive functionals on the spaces Lp, 0 < p < 1, are presented. The convergence of additive functionals is also investigated whenever three types of convergence are imposed on the members of Lp: almost everywhere convergence, convergence in measure, and convergence in the metric d, where d(x,y) = |xy|p . In all three cases an integral representation for the functional is obtained, and necessary and sufficient conditions are given for the continuity of the functional.

Mathematical Subject Classification 2000
Primary: 46E30
Milestones
Received: 23 September 1975
Revised: 3 June 1976
Published: 1 September 1976
Authors
Judith Ann Palagallo