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Let E be a Banach function space and let X be a real Banach space. We examine weakly compact linear operators from a Köthe-Bochner space E(X) endowed with some natural mixed topology ${\gamma }_{E\left(X\right)}$ (in the sense of Wiweger) to a Banach space Y.

We examine the topological properties of Orlicz-Bochner spaces $L^{\varphi }\left(X\right)$ (over a σ-finite measure space $(\Omega ,\Sigma ,\mu ))$, where $\varphi$ is an Orlicz function (not necessarily convex) and $X$ is a real Banach space. We continue the study of some class of locally convex topologies on $L^{\varphi }\left(X\right)$, called uniformly $\mu$-continuous topologies. In particular, the generalized mixed topology ${𝒯}_I^{\varphi }\left(X\right)$ on $L^{\varphi }\left(X\right)$ (in the sense of Turpin) is considered.

Let $E$ be an ideal of $L^0$ over $\sigma$-finite measure space $(\Omega ,\Sigma ,\mu )$ and let $(X,\parallel ·{\parallel }_X)$ be a real Banach space. Let $E\left(X\right)$ be a subspace of the space $L^0\left(X\right)$ of $\mu$-equivalence classes of all strongly $\Sigma$-measurable functions $f:\Omega \to X$ and consisting of all those $f\in L^0\left(X\right)$, for which the scalar function $\tilde{f}={\parallel f(·)\parallel }_X$ belongs to $E$. Let $E$ be equipped with a Hausdorff locally convex-solid topology $\xi$ and let $\xi$ stand for the topology on $E\left(X\right)$ associated with $\xi$. We examine the relationship between the properties of the space $\left(E\right(X),\xi )$ and the properties of both the spaces $(E,\xi )$ and $(X,\parallel ·{\parallel }_X)$....

Some class of locally solid topologies (called uniformly $\mu$-continuous) on Köthe-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 ${𝒯}_I^{\varphi }\left(X\right)$ on the Orlicz-Bochner space $L^{\varphi }\left(X\right)$ is a generalized mixed topology in the sense of P. Turpin (see [11, Chapter I]).

Locally solid topologies on vector valued function spaces are studied. The relationship between the solid and topological structures of such spaces is examined.