Sequential cone-compactness does not imply cone-compactness
Marius Durea, Elena-Andreea Florea
TL;DR
This paper resolves the question of whether sequential cone compactness implies cone compactness without assuming separability by constructing a nonseparable normed space example. The authors define two cone-based compactness notions, show the known implication $C$-compactness $\Rightarrow$ $C$-sequentially compactness, and recall that separability suffices for the converse. They then exhibit a concrete $A \subset B(\mathbb{R})$ with cone $C = B(\mathbb{R})_+$ that is not $C$-compact but is $C$-sequentially compact, demonstrating that sequential cone compactness is strictly weaker than cone compactness beyond separability. This clarifies the limits of using sequential cone compactness as a surrogate for cone compactness in the nonseparable setting and informs related results in the literature.
Abstract
We address a problem posed in [1] by demonstrating through an example that, in the absence of separability, the property of sequential cone compactness does not generally imply cone compactness.