The space of all order continuous linear functionals on an Orlicz space $L^{\varphi }$ defined by an arbitrary (not necessarily convex) Orlicz function $\varphi$ is described.

We study order convexity and concavity of quasi-Banach Lorentz spaces ${\Lambda }_{p,w}$, where 0 < p < ∞ and w is a locally integrable positive weight function. We show first that ${\Lambda }_{p,w}$ contains an order isomorphic copy of $l^p$. We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for ${\Lambda }_{p,w}$. We conclude with a characterization of the type and cotype of ${\Lambda }_{p,w}$ in the case when ${\Lambda }_{p,w}$ is a normable space.

Let $K$ be a compact space and let $C\left(K\right)$ be the Banach lattice of real-valued continuous functions on $K$. We establish eleven conditions equivalent to the strong compactness of the order interval $[0,x]$ in $C\left(K\right)$, including the following ones: (i) $\{x>0\}$ consists of isolated points of $K$; (ii) $[0,x]$ is pointwise compact; (iii) $[0,x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0,x]$; (v) the strong and weak topologies coincide on $[0,x]$. Moreover, the weak topology and that of pointwise convergence...

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We show that $E\left(\mu \right)$ is order bounded if and only if it is contained in a principal ideal in $ba\left(\Re \right)$ if and only if it is weakly compact and $extrE\left(\mu \right)$ is contained in a principal ideal in $ba\left(\Re \right)$. We also establish some criteria for the coincidence of the ideals, in $ba\left(\Re \right)$, generated by $E\left(\mu \right)$ and $extrE\left(\mu \right)$.