Let ℬ be an ordered normed
space with positive cone ℬ+ and let N be the canonical half-norm of ℬ+,
i.e.,
Then, for any hereditary subcone 𝒞 of ℬ+, the positive bipolar 𝒞⊥⊥ coincides with
the N-closure 𝒞N of 𝒞, i.e.,
Similar facts are proved for order ideals and these results are used to derive a result
of Størmer on archimedean order ideals.
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