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Let $\alpha>0$ and $X$ be an $\alpha$-normal (meaning, for $x,y\in X$, $0\leq x\leq y$ implies $\|x\|\leq\alpha\|y\|$) ordered Banach space with closed generating cone $X_{+}$. If $X$ is reflexive, th …

The answer is that the reflexivity assumption cannot be dropped. The following simple example (due to Tony Wickstead) is a 1-normal non-reflexive space with closed and generating cone, where there ex …

I feel like I should add that there exists a nice characterization due to Tony Wickstead of ordered Banach spaces whose compact sets are order bounded. Wickstead A.W., Compact subsets of partially or …

I had the following little question pop up, but I cannot seem to find any reference to it. Let $X$ be a Banach space and $E\subseteq X$ a proper subspace with $E$ isomorphic to $X$ itself. Is th …

Let $X$ be a Banach space and $Y$ a closed subspace of $X$. I am interested in quotients $q:X\to X/Y$ that do not have Lipschitz right inverses (not necessarily linear). Of course, if $Y$ is compleme …

Let $ces_0:=\{\xi\in\ell^\infty : \lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^{n}\xi_k=0\}$ and $q:\ell^\infty \to \ell^\infty/ces_0$ be the usual quotient map. The space $ces_0$ is closed in $\ell^\inft …

If $X$ is a Real Banach space with strictly convex norm, it is known that for any non-empty compact, convex set $K$ and point $x_0\notin K$, there exists a unique point $z_0\in K$ minimizing the quant …

After some digging, I found these two papers on the subject: Batty, Charles, and Derek Robinson. “Positive One-parameter Semigroups on Ordered Banach Spaces.” Acta Applicandae Mathematicae 2, no. 3 …

I'm trying to investigate the interplay between the norm and cone of positive elements in ordered Banach spaces. In particular, I would like a nice characterization of when the norm of a positive oper …

$\DeclareMathOperator\Lip{Lip}$Let $X$ be a real Banach space. The dual $X^*$ is a closed subspace of $\Lip_0(X)$. ($\Lip_0(X)$ denotes the space of real-valued Lipschitz functions $f:X\to\mathbb{R}$ …