Basic topological and geometric properties of Ces\`{a}ro--Orlicz spaces

Necessary and sufficient conditions under which the Ces\`{a}ro--Orlicz sequence space $\cfi$ is nontrivial are presented. It is proved that for the Luxemburg norm, Ces\`{a}ro--Orlicz spaces $\cfi$ have the Fatou property. Consequently, the spaces are complete. It is also proved that the subspace of order continuous elements in $\cfi$ can be defined in two ways. Finally, criteria for strict monotonicity, uniform monotonicity and rotundity (= strict convexity) of the spaces $\cfi$ are given.