Abstract:Some class of locally solid topologies (called uniformly $\mu $-continuous) on K\"othe-Bochner spaces that are continuous with respect to some natural two-norm convergence are introduced and studied. A characterization of uniformly $\mu $-continuous topologies in terms of some family of pseudonorms is given. The finest uniformly $\mu $-continuous topology $\Cal T^\varphi _I(X)$ on the Orlicz-Bochner space $L^\varphi (X)$ is a generalized mixed topology in the sense of P. Turpin (see [11, Chapter I]).
Keywords: Orlicz spaces, Orlicz-Bochner spaces, K\"othe-Bochner spaces, locally solid topologies, generalized mixed topologies, uniformly $\mu $-continuous topologies, inductive limit topologies
AMS Subject Classification: 46E30, 46E40, 46A70