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vector valued function spaces; locally solid topologies; strong topologies; Mackey topologies; absolute weak topologies
Let $(X,\Vert \cdot \Vert _X)$ be a real Banach space and let $E$ be an ideal of $L^0$ over a $\sigma $-finite measure space $(Ø,\Sigma ,\mu )$. Let $(X)$ be the space of all strongly $\Sigma $-measurable functions $f\: Ø\rightarrow X$ such that the scalar function ${\widetilde{f}}$, defined by ${\widetilde{f}}(ø)=\Vert f(ø)\Vert _X$ for $ø\in Ø$, belongs to $E$. The paper deals with strong topologies on $E(X)$. In particular, the strong topology $\beta (E(X), E(X)^\sim _n)$ ($E(X)^\sim _n=$ the order continuous dual of $E(X)$) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.
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