### Linear functionals on Orlicz sequence spaces without local convexity.

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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 $(X,|\left|\xb7\right|{|}_{X})$ 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 ${\left|\right|f\left(\xb7\right)\left|\right|}_{X}$ belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let ${D}_{u}(={f\in E\left(X\right):\left|\right|f\left(\xb7\right)\left|\right|}_{X}\le u)$ stand for the order interval in E(X). For a real Banach space $(Y,|\left|\xb7\right|{|}_{Y})$ 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 $(\tau ({L}^{\infty}\left(X\right),L\xb9(X*)),|\left|\xb7\right|{|}_{Y})$-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 ${i}_{\infty}:{L}^{\infty}\left(X\right)\to L\xb9\left(X\right)$ 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 $(\beta ,|\left|\xb7\right|{|}_{F})$-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 $(X,\parallel \xb7{\parallel}_{X})$ be a real Banach space and let $E$ be an ideal of ${L}^{0}$ over a $\sigma $-finite measure space $(\xd8,\Sigma ,\mu )$. Let $\left(X\right)$ be the space of all strongly $\Sigma $-measurable functions $f\phantom{\rule{0.222222em}{0ex}}\xd8\to X$ such that the scalar function $\tilde{f}$, defined by $\tilde{f}\left(\xf8\right)={\parallel f\left(\xf8\right)\parallel}_{X}$ for $\xf8\in \xd8$, 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.

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 $(Y,\parallel \xb7{\parallel}_{Y})$. In particular, we derive Banach-Steinhaus type theorems for $({\beta}_{z},\parallel \xb7{\parallel}_{Y})$ 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 $Y$.

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 ${L}^{\varphi}$ 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 ${\mathcal{T}}_{\varphi}$ 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...

We study linear operators from a non-locally convex Orlicz space ${L}^{\Phi}$ to a Banach space $(X,|\left|\xb7\right|{|}_{X})$. Recall that a linear operator $T:{L}^{\Phi}\to X$ is said to be σ-smooth whenever $u\u2099{\u27f6}^{\left(o\right)}0$ in ${L}^{\Phi}$ implies ${\left|\right|T\left(u\u2099\right)\left|\right|}_{X}\to 0$. It is shown that every σ-smooth operator $T:{L}^{\Phi}\to X$ factors through the inclusion map $j:{L}^{\Phi}\to {L}^{\Phi \u0305}$, 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...

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

Let $X$ be a completely regular Hausdorff space and $E$ a real normed space. We examine the general properties of locally solid topologies on the space ${C}_{b}(X,E)$ of all $E$-valued continuous and bounded functions from $X$ into $E$. The mutual relationship between locally solid topologies on ${C}_{b}(X,E)$ and ${C}_{b}\left(X\right)$ $(={C}_{b}(X,\mathbb{R}))$ is considered. In particular, the mutual relationship between strict topologies on ${C}_{b}\left(X\right)$ and ${C}_{b}(X,E)$ is established. It is shown that the strict topology ${\beta}_{\sigma}(X,E)$ (respectively ${\beta}_{\tau}(X,E)$) is the finest $\sigma $-Dini topology (respectively...

Let $X$ be a completely regular Hausdorff space, $E$ a real normed space, and let ${C}_{b}(X,E)$ be the space of all bounded continuous $E$-valued functions on $X$. We develop the general duality theory of the space ${C}_{b}(X,E)$ endowed with locally solid topologies; in particular with the strict topologies ${\beta}_{z}(X,E)$ for $z=\sigma ,\tau ,t$. As an application, we consider criteria for relative weak-star compactness in the spaces of vector measures ${M}_{z}(X,{E}^{\text{'}})$ for $z=\sigma ,\tau ,t$. It is shown that if a subset $H$ of ${M}_{z}(X,{E}^{\text{'}})$ is relatively $\sigma ({M}_{z}(X,{E}^{\text{'}}),{C}_{b}(X,E))$-compact, then the set $conv\left(S\right(H\left)\right)$ is still relatively $\sigma ({M}_{z}(X,{E}^{\text{'}}),{C}_{b}(X,E))$-compact...

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