Consider a real, normed vector Banach$^1$ space $V$. We'll call a subset $V_+ \subseteq V$ a cone if
Is it true that a solid, minihedral cone in an infinite-dimensional, real normedBanach space cannot be regular?
Remark: The cones in (1) and (2) above are also normal---meaning that there exists a $C \geqslant 0$ such that $0 \leqslant x \leqslant y$ implies $\|x\| \leqslant C\|y\|$. In fact, $C = 1$. But that's not the problem since regularity actually implies normality in Banach spaces [1, Lemma 5.1 and Theorem 5.1].
In particular, $(w_n)$ is order-bounded by $u$, so it converges in norm by regularity. I haven't yet been able to see the contradiction here though.
(4) I might have found a solution using a representation theorem due to Kakutani.
Theorem (Kakutani) If $V$ is a real Banach space, partially ordered by a solid, normal, minihedral cone $V_+$, then there exists a compact, Hausdorff space $K$ and a linear homeomorphism $\Phi\colon V \rightarrow C(K)$ such that $\Phi(V_+) = C_+(K)$, where $C_+(K)$ is the cone of nonnegative, continuous functions in $C(K)$.
Proof See [1, Theorem 6.6].
Since regularity implies normality, a solid, minihedral, regular cone in a real Banach space satisfies the hypotheses of the theorem. Since $\Phi$ preserves solidity, minihedrality and regularity, it's enough to prove the result for $C(K)$. Now $C(K)$ is finite-dimensional if, and only if $K$ is finite. I showed that if $K$ is infinite, then we can essentially find a copy of $l^\infty$ in $C(K)$, contradicting the hypothesis that $C(K)$ is regular (Example (1) above). Thus if $C(K)$ is regular, then $K$ must be finite.
I'm still checking all the details though, it's quite an elaborate construction, and depends on this nontrivial representation result of Kakutani. So I'm not too happy about it. Since it's an exercise in the book I thought there must have been a simpler solution. Plus, the exercise is given before Kakutani's theorem. So I imagine the authors thought it could be solved without it?
Footnotes
- Completeness is assumed throughout [1], seems to be needed even for basic properties such as characterizations of normality, and is present in all applications I've come across so far. So I'm OK with assuming that $V$ is Banach.
References
2013-03-27: Added $l^p$ example and remark about normality.
2013-04-04: Added completeness to the hypotheses in $V$, observation that regularity implies normality, and the representation theorem of Kakutani.