Lebesgue-Bochner Function Spaces Research Articles

L0-Convex Compactness and Random Normal Structure in L0 (F, B)

Let (B, ∥ · ∥) be a Banach space, (Ω, F, P) a probability space, and L0 (F, B) the set of equivalence classes of strong random elements (or strongly measurable functions) from (Ω, F, P) to (B, ∥ · ∥). It is well known that L0 (F, B) becomes a complete random normed module, which has played an important role in the process of applications of random normed modules to the theory of Lebesgue-Bochner function spaces and random operator theory. Let V be a closed convex subset of B and L0 (F, V) the set of equivalence classes of strong random elements from (Ω, F, P) to V. The central purpose of this article is to prove the following two results: (1) L0 (F, V) is L0-convexly compact if and only if V is weakly compact; (2) L0 (F, V) has random normal structure if V is weakly compact and has normal structure. As an application, a general random fixed point theorem for a strong random nonexpansive operator is given, which generalizes and improves several well known results. We hope that our new method, namely skillfully combining measurable selection theorems, the theory of random normed modules, and Banach space techniques, can be applied in the other related aspects.

Hermite-Hadamard's inequality and the p-HH-norm on the Cartesian product of two copies of a normed space

The Cartesian product of two copies of a normed space is naturally equipped with the well-known p -norm. In this paper, another notion of norm is introduced, and will be called the p - HH -norm. This norm is an extension of the generalised logarithmic mean and is connected to the p -norm by the Hermite-Hadamard’s inequality. The Cartesian product space (with respect to both norms) is complete, when the (original) normed space is. A proof for the completeness of the p - HH -norm via Ostrowski’s inequality is provided. This space is embedded as a subspace of the well-known Lebesgue-Bochner function space (as a closed subspace, when the norm is a Banach norm). Consequently, its geometrical properties are inherited from those of Lebesgue-Bochner space. An explicit expression of the superior (inferior) semi-inner product associated to both norms is considered and used to provide alternative proofs for the smoothness and reflexivity of this space.

Representation theorems of the dual of Lebesgue-Bochner function spaces

By representing random conjugate spaces a general representation theorem on classical duals is proved. for application,we unify and improve many known imjportant representation theorems of the dual of Lebesgue-Bichner function spaces.

The near Radon-Nikodym property in Lebesgue-Bochner function spaces

Let $X$ be a Banach space $E$ a Kothe function space that does not contain $c_0$. It is shown that the vector valued function space $E(X)$ has the Near Radon Nikodym property if and only if $X$ does.

Complex convexity in Lebesgue-Bochner Function Spaces

Complex geometric properties of continuously quasi-normed spaces are introduced and their relationship to their analogues in real Banach spaces is discussed. It is shown that these properties lift from a continuously quasi-normed space X X to L p ( μ , X ) L^p(\mu , X) , for 0 > p > ∞ 0 > p > \infty . Local versions of these properties and results are also considered.

Strongly Exposed Points in Lebesgue-Bochner Function Spaces

It is a result of Peter Greim that if $f$ is a strongly exposed point of the unit ball of Lebesgue-Bochner function space ${L^p}(\mu ,X),\;1 < p < \infty$, then $f$ is a unit vector and $f(t)/||f(t)||$ is a strongly exposed point of the unit ball of $X$ for almost all $t$ in the support of $f$. We prove that the converse is also true.

Asymptotic-norming and Mazur intersection properties in Bochner function spaces

A Banach space X has the asymptotic-norming property if and only if the Lebesgue-Bochner function space Lp (μ, X) has the asumptotic-norming property for p with 1 &lt; p &lt; ∞. It follows that a Banach space X is Hahn-Banach smooth if and only if Lp (μ, X) is Hahn-Banach smooth for p with 1 &lt; p &lt; ∞. We also show that for p with 1 &lt; p &lt; ∞, (1) if X has the compact Mazur intersection property then so does Lp(μ, X); (2) if the measure μ is not purely atomic, then the space Lp(μ, X) has the Mazur intersection property if and only if X is an Asplund space and has the Mazur intersection property.

A note on proximinality in function spaces F(S,X)

We discuss initially how to construct a generalisation of the Lebesgue-Bochner function spaces by using a Banach lattice of real-valued functions and an appropriate lifting of the norm from this space to the more general setting. This lifting is shown to be well-defined, but the primary purpose is to discuss the notion of proximinality in this new space. In this paper, the results are derived by establishing the property of uniform convexity under suitable hypotheses. Other techniques may be found in the literature, and citations are given in the paper.

Rotundity in Lebesgue-Bochner function spaces

This paper concerns the isometric theory of the Lebesgue-Bochner function space L p ( μ , X ) {L^p}(\mu ,\,X) where 1 &gt; p &gt; ∞ 1 &gt; p &gt; \infty . Specifically, the question of whether a geometrical property lifts from X to L p ( μ , X ) {L^p}\,(\mu ,\,X) is examined. Positive results are obtained for the properties local uniform rotundity, weak uniform rotundity, uniform rotundity in each direction, midpoint local uniform rotundity, and B-convexity. However, it is shown that the Radon-Riesz property does not lift from X to L p ( μ , X ) {L^p}\,(\mu ,\,X) . Consequently, Lebesgue-Bochner function spaces with the Radon-Riesz property are examined more closely.

Rotundity in Lebesgue-Bochner Function Spaces

This paper concerns the isometric theory of the Lebesgue-Bochner function space ${L^p}(\mu , X)$ where $1 < p < \infty$. Specifically, the question of whether a geometrical property lifts from

The Radon-Nikodym theorem for Lebesgue-Bochner function spaces

It is proved that the Lebesgue-Bochner function spaces L p ( E, λ), 1 < p < ∞, have the Radon-Nikodym property if and only if E has the same property. This solves an unsolved problem of Davies, Diestel and Leonard.

Geometry of Lebesgue-Bochner function spaces―smoothness

There exist real Banach spaces $E$ such that the norm in $E$ is of class ${C^\infty }$ away from zero; however, for any $p,1 \leqslant p \leqslant \infty$, the norm in the Lebesgue-Bochner function space ${L_p}(E,\mu )$ is not even twice differentiable away from zero. The main objective of this paper is to give a complete determination of the order of differentiability of the norm function in this class of Banach spaces.

Geometry of Lebesgue-Bochner function spaces—smoothness

There exist real Banach spaces E such that the norm in E is of class C' away from zero; however, for any p, I < p < oo, the norm in the Lebesgue-Bochner function space LP(E, ,u) is not even twice differentiable away from zero. The main objective of this paper is to give a complete determination of the order of differentiability of the norm function in this class of Banach spaces. Introduction. The class of Lebesgue-Bochner function spaces, introduced by Bochner and Taylor [4] in 1938, has been found to be of considerable importance in various branches of mathematics, and is discussed at length in Dinculeanu [11], Dunford and Schwartz [12], and Edwards [13]. The study of the geometric properties of the Lebesgue-Bochner function spaces dates back about three decades: Day [8] and McShane [17], respectively, characterized uniform convexity and smoothness of these spaces. In fact, the only known result concerning the smoothness of the Lebesgue-Bochner function spaces is due to McShane, and his result concerns only the directional derivative (Gateaux derivative) of the norm in this class of Banach spaces. Even the Frechet differentiability of the norm has not been considered anywhere. It might be mentioned in this connection that the first systematic study of higher-order differentiability of the norm in a Banach space was made by Kurzweil [15] in 1954. Subsequently, in 1965, Bonic and Frampton [5a] extended Kurzweil's results, and in 1966 [5b] they discussed various categories of smooth Banach manifolds. In 1967, Sundaresan [18] extended some of Kurzweil's results independently. In [5b] and [18], the order of differentiability of the norm in the classical Lp spaces, 1 < p < oo, is obtained; while in Sundaresan [20], the smoothness of the norm in C(X, E) is discussed. For an elegant up-to-date account of smooth Banach spaces, and related concepts one might refer to the lecture notes by S. Yamamuro [23]. This paper contains the first systematic investigation of the higher-order differentiability of the norm function in the LebesgueBochner function spaces Received by the editors April 28, 1973. AMS (MOS) subject classifications (1970). Primary 46E40, 28A45; Secondary 58C20, 28A15.

Representation of Vector Valued Nonlinear Functions

A representation theorem for “additive” nonlinear functional on spaces ${L^p}(\mu )$ is here extended to “additive” nonlinear functions from Lebesgue-Bochner function spaces $L_E^p(\mu )$ ($E$ a separable Banach space) into Banach spaces $B$. A counterexample is provided to show that the restriction to separable $E$ is essential.

Errata: Extreme Points of the Unit Cell in Lebesgue-Bochner Function Spaces

Errata: Extreme Points of the Unit Cell in Lebesgue-Bochner Function Spaces

Extreme Points of the Unit Cell in Lebesgue-Bochner Function Spaces. I

Extreme Points of the Unit Cell in Lebesgue-Bochner Function Spaces. I