Vol. 110, No. 2, 1984

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Hereditary cones, order ideals and half-norms

Derek W. Robinson and Sadayuki Yamamuro

Vol. 110 (1984), No. 2, 335–343
Abstract

Let be an ordered normed space with positive cone + and let N be the canonical half-norm of +, i.e.,

N(a) = inf{∥a+ b∥;b ∈ ℬ+}.

Then, for any hereditary subcone 𝒞 of +, the positive bipolar 𝒞⊥⊥ coincides with the N-closure 𝒞N of 𝒞, i.e.,

𝒞N = {a ∈ ℬ+;N (a− an) → 0 for some an ∈ 𝒞}.

Similar facts are proved for order ideals and these results are used to derive a result of Størmer on archimedean order ideals.

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