Vol. 110, No. 2, 1984

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The Jordan decomposition and half-norms

Derek W. Robinson and Sadayuki Yamamuro

Vol. 110 (1984), No. 2, 345–353
Abstract

Let be a Banach space, with norm ∥⋅∥, ordered by a positive cone + and order the dual by the dual cone +. We prove that, if is orthogonally generated, each f ∈ℬ has an orthogonal, and norm-unique, Jordan decomposition f = f+ f with f±∈ℬ,

∥f∥ = ∥f+∥+ ∥f− ∥,

if, and only if, the norm on has the order theoretic property

∥a∥ = inf{λ ≥ 0;− λu ≤ a ≤ λv for some u,v ∈ ℬ1},

when 1 is the unit ball of . Various characterizations of the canonical half-norm associated with + are also given.

Mathematical Subject Classification 2000
Primary: 46B30, 46B30
Secondary: 46A40
Milestones
Received: 13 May 1982
Published: 1 February 1984
Authors
Derek W. Robinson
Sadayuki Yamamuro