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If $V$ is a normed vector space then the natural map from $V$ to its double dual $V''$ is norm-preserving as follows from Hahn-Banach theorem. This is well-known.

Now assume that P is just a vector cone, which is equipped with a norm satisfying usual axioms, plus monotonicity: if $a\le b$ then $\|a\|\le\|b\|$. (The standard order on $P$ is defined by $a\le b$ if there exists $x$ such that $b=x+a$.)

We have the dual cone $P'$ of norm-bounded positive functionals on P and the dual norm $\|p\|=\sup\{(p,a):\|a\|\le 1,a\ge 0\}$.

Then we have a natural map from $P$ to $P''$, which is obviously norm-bounded. Is anything known on the conditions for it to be norm-preserving, just as in the case of vector spaces?

I was unable to find any hints in the literature. Although the question seems quite natural.

(We can always think of $P$ as embedded in the enveloping vector space $V=P-P$, which becomes partially ordered by $P$, and the norm extends from $P$ to $V$ by $\|b\|=\inf\{\|a\|: -a\le b\le a\}$ to a monotone norm, i.e satisfying the condition if $-a\le b\le a$ then $\|b\|\le\|a\|$. But I am not sure that this is important.)

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I settled this, the answer is: always.

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    $\begingroup$ It would generally be better to post the proof; you are allowed (and encouraged) to answer your own questions when you think you have the answer. $\endgroup$
    – user44191
    Commented Mar 16, 2020 at 19:05
  • $\begingroup$ Hahn-Banach theorem extension theorem cen be adapted to cones with monotone norm. I.e. a bounded positive functional defined on a subcone of a normed cone extends to a positive funnctional on the whole cone of the same norm. (That the norm is monotone is important!!) A proof is very similar to the standard Hahn-Banach for vector spaces. I'll, hopefully post it, when I have more time. $\endgroup$
    – Sergey Slavnov
    Commented Mar 16, 2020 at 20:25

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