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Let L-phi be an Orlicz space defined by a Young function phi over a sigma-finite measure space, and let phi* denote the complementary function in the sense of Young. We give a characterization of the Mackey topology tau(L*,L-phi*) in terms of some family of norms defined by some regular Young functions. Next we describe order continuous (=absolutely continuous) Riesz seminorms on L-phi, and obtain a criterion for relative sigma(L-phi,L-phi*)-compactness in L-phi. As an application we get a representation...

Let $X$ be a completely regular Hausdorff space, $E$ a real Banach space, and let $C_b(X,E)$ be the space of all $E$-valued bounded continuous functions on $X$. We study linear operators from $C_b(X,E)$ endowed with the strict topologies ${\beta }_z$ $(z=\sigma ,\tau ,\infty ,g)$ to a real Banach space $(Y,\parallel ·{\parallel }_Y)$. In particular, we derive Banach-Steinhaus type theorems for $({\beta }_z,\parallel ·{\parallel }_Y)$ continuous linear operators from $C_b(X,E)$ to $Y$. Moreover, we study $\sigma$-additive and $\tau$-additive operators from $C_b(X,E)$ to $Y$.

Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T : Cb(X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · ||F)-continuous operators T : Cb(X, E) → F. As an application, we...

Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space $(X,|\left|·\right|{|}_X)$ let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function ${\left|\right|f\left(·\right)\left|\right|}_X$ belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let $D_u(={f\in E\left(X\right):\left|\right|f\left(·\right)\left|\right|}_X\le u)$ stand for the order interval in E(X). For a real Banach space $(Y,|\left|·\right|{|}_Y)$ a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈ E⁺ the set...

Let (Ω,Σ,μ) be a finite measure space and let X be a real Banach space. Let $L^{\Phi }\left(X\right)$ be the Orlicz-Bochner space defined by a Young function Φ. We study the relationships between Dunford-Pettis operators T from L¹(X) to a Banach space Y and the compactness properties of the operators T restricted to $L^{\Phi }\left(X\right)$. In particular, it is shown that if X is a reflexive Banach space, then a bounded linear operator T:L¹(X) → Y is Dunford-Pettis if and only if T restricted to $L^{\infty }\left(X\right)$ is $(\tau (L^{\infty }\left(X\right),L¹(X*)),|\left|·\right|{|}_Y)$-compact.

A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let $i_{\infty }:L^{\infty }\left(X\right)\to L¹\left(X\right)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T\circ i_{\infty }:L^{\infty }\left(X\right)\to Y$ is a weakly compact operator. Moreover, we obtain that if T: L¹(X)...

Let X be a completely regular Hausdorff space, E and F be Banach spaces. Let $C_b(X,E)$ be the space of all E-valued bounded continuous functions on X, equipped with the strict topology β. We study topological properties of the space $L_{\beta }(C_b(X,E),F)$ of all $(\beta ,|\left|·\right|{|}_F)$-continuous linear operators from $C_b(X,E)$ to F, equipped with the topology ${\tau }_s$ of simple convergence. If X is a locally compact paracompact space (resp. a P-space), we characterize ${\tau }_s$-compact subsets of $L_{\beta }(C_b(X,E),F)$ in terms of properties of the corresponding sets of the representing...

Let $\Sigma$ be a $\sigma$-algebra of subsets of a non-empty set $\Omega$. Let $ba\left(\Sigma \right)$ be the complex vector lattice of bounded finitely additive measures $\mu :\Sigma \to ℂ$. We study locally solid topologies on $ba\left(\Sigma \right)$. We develop the duality theory of $ba\left(\Sigma \right)$, provided with locally convex-solid topologies.

Let $E$ be a Banach function space and $X$ be a real Banach space. We study Bochner representable operators from a Köthe-Bochner space $E\left(X\right)$ to a Banach space $Y$. We consider the problem of compactness and weak compactness of Bochner representable operators from $E\left(X\right)$ (provided with the natural mixed topology) to $Y$.

Let $L^{\varphi }$ be an Orlicz space defined by a convex Orlicz function $\varphi$ and let $E^{\varphi }$ be the space of finite elements in $L^{\varphi }$ (= the ideal of all elements of order continuous norm). We show that the usual norm topology ${𝒯}_{\varphi }$ on $L^{\varphi }$ restricted to $E^{\varphi }$ can be obtained as an inductive limit topology with respect to some family of other Orlicz spaces. As an application we obtain a characterization of continuity of linear operators defined on $E^{\varphi }$.

Let $L^{\varphi }\left(X\right)$ be an Orlicz-Bochner space defined by an Orlicz function $\varphi$ taking only finite values (not necessarily convex) over a $\sigma$-finite atomless measure space. It is proved that the topological dual $L^{\varphi }{\left(X\right)}^{*}$ of $L^{\varphi }\left(X\right)$ can be represented in the form: $L^{\varphi }{\left(X\right)}^{*}=L^{\varphi }{\left(X\right)}_n^{\sim }\oplus L^{\varphi }{\left(X\right)}_s^{\sim }$, where $L^{\varphi }{\left(X\right)}_n^{\sim }$ and $L^{\varphi }{\left(X\right)}_s^{\sim }$ denote the order continuous dual and the singular dual of $L^{\varphi }\left(X\right)$ respectively. The spaces $L^{\varphi }{\left(X\right)}^{*}$, $L^{\varphi }{\left(X\right)}_n^{\sim }$ and $L^{\varphi }{\left(X\right)}_s^{\sim }$ are examined by means of the H. Nakano’s theory of conjugate modulars. (Studia Mathematica 31 (1968), 439–449). The well known results of the duality theory...

The space of all order continuous linear functionals on an Orlicz space $L^{\varphi }$ defined by an arbitrary (not necessarily convex) Orlicz function $\varphi$ is described.

Let $(X,\parallel ·{\parallel }_X)$ be a real Banach space and let $E$ be an ideal of $L^0$ over a $\sigma$-finite measure space $(Ø,\Sigma ,\mu )$. Let $\left(X\right)$ be the space of all strongly $\Sigma$-measurable functions $fØ\to X$ such that the scalar function $\tilde{f}$, defined by $\tilde{f}\left(ø\right)={\parallel f\left(ø\right)\parallel }_X$ for $ø\in Ø$, belongs to $E$. The paper deals with strong topologies on $E\left(X\right)$. In particular, the strong topology $\beta (E\left(X\right),E{\left(X\right)}_n^{\sim })$ ($E{\left(X\right)}_n^{\sim }=$ the order continuous dual of $E\left(X\right)$) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.

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

We study linear operators from a non-locally convex Orlicz space $L^{\Phi }$ to a Banach space $(X,|\left|·\right|{|}_X)$. Recall that a linear operator $T:L^{\Phi }\to X$ is said to be σ-smooth whenever $uₙ{⟶}^{\left(o\right)}0$ in $L^{\Phi }$ implies ${\left|\right|T\left(uₙ\right)\left|\right|}_X\to 0$. It is shown that every σ-smooth operator $T:L^{\Phi }\to X$ factors through the inclusion map $j:L^{\Phi }\to L^{\Phi ̅}$, where Φ̅ denotes the convex minorant of Φ. We obtain the Bochner integral representation of σ-smooth operators $T:L^{\Phi }\to X$. This extends some earlier results of J. J. Uhl concerning the Bochner integral representation of linear operators defined on a locally convex...