Background

Consider a real, normed vector space $V$. We'll call a subset $V_+ \subseteq V$ a cone if

• $V$ is closed,
• $\alpha V_+ \subseteq V_+$ for every $\alpha \geqslant 0$ and
• $V_+ \cap (-V_+) = \{0\}$.

Cones induce a partial order $\leqslant$ in $V$:

$$x \leqslant y \quad \Longleftrightarrow \quad y - x \in V_+.$$

A cone is said to be solid if it has nonempty interior $intV_+ \neq \varnothing$. It is said to be minihedral if every finite subset $B$ of $V$ has a supremum (least upper bound). It is said to be regular if every order-bounded, monotone sequence converges in norm:

$$x_1 \leqslant x_2 \leqslant \cdots \leqslant x_n \leqslant \cdots \leqslant u \quad \Longrightarrow \quad x_n \longrightarrow x_\infty.$$

I'm interested in cone-induced partial orders in the context of Monotone Dynamical Systems. Properties such as the ones listed above often come up in the discussion of asymptotic behavior.

Question

Is it true that a solid, minihedral cone in an infinite-dimensional real normed space cannot be regular?

This is listed as Exercise 6.7 on page 61 in [1]. However I've found statements in exercises in this book to be false---in fact, the very first exercise is false---and there are many other mistakes such as theorems without all hypotheses listed. So I'm skeptical.

Progress

(1) The set $l^\infty_+$ of nonnegative, bounded sequences is a solid and minihedral cone in $l^\infty$. But it is not regular. Setting

$$x_n := (1, 1, 1, \ldots, 1, 0, 0, 0, \ldots) \quad \text{($n$ $1$'s)}$$

we have a monotone sequence order-bounded from above by $(1, 1, 1, \ldots)$ which does not converge in norm.

(2) I've tried to prove it by contradiction, assuming that the cone is solid, minihedral and regular in an infinite-dimensional space. I picked a sequence of unit vectors $(v_n)$ such that

$$dist(v_{n+1}, span \{ v_1,\ldots,v_n \} ) \geqslant 1/2.$$

Because the cone is solid, there exists an $u \in V$ such that

$$\{v_n\} \subseteq B_1(0) \subseteq [-u, u] := \{x \in V\,;\ -u \leqslant x \leqslant u\}.$$

By the minihedrality assumption, a monotone sequence $(w_n)_{n \in \mathbb{N}}$ is well-defined by

$$w_n := \sup\{v_1,\ldots,v_n\}.$$

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.

References

[1] Krasnosel'skij, Lifshits and Sobolev; Positive Linear Systems---The Method of Positive Operators. Heldermann Verlag Berlin, 1989.