A Counterexample to Two Compactness Conjectures of Brudnyi

We construct a quasi-Banach function lattice $X$ on $\mathbb{R}$ whose quasi-norm is order continuous and invariant under every translation. Nevertheless, $X$ contains a bounded sequence with common compact support which is equicontinuous in $X$ and has no norm-convergent subsequence. This provides a counterexample to Conjectures 2.5 and 2.11 proposed by Brudnyi in \cite{Brudnyi2015}.