Two-Norm Convergence in the Lp Spaces

Abstract

We have characterized the spaces $X$ for which the smallest $z$-ideal containing $c_\infty$ is prime. It turns out that $c_\infty$ is a $z$-ideal in $C(X)$ if and only if every zero-set contained in an open locally compact $\sigma$-compact set is compact. Some interesting ideals related to $c_\infty$ are introduced and corresponding to the relations between these ideals and $c_\infty$, topological spaces $X$ are characterized. Some compactness concepts are explicitly stated in terms of ideals related to $c_\infty$. Finally we have shown that a $\sigma$-compact space $X$ is Baire if and only if every ideal containing $c_\infty$ is essential.