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Register a new userA Local Hahn-Banach Theorem and Its Applications
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An important consequence of the Hahn-Banach Theorem says that on any locally convex Hausdorff topological space $X$, there are sufficiently many continuous linear functionals to separate points of $X$. In the paper, we establish a `local version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in [3]. We also provide a simplified approach to the measure-free characterization of uniform integrability established in [8].
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[REVISED VERSION] The aim of this paper is to state a sharp version of the Konig supremum theorem, an equivalent reformulation of the Hahn--Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush-Kuhn-Tucker and Fritz John type, for nonlinear infinite programs. We also show that a weak concept of convexity coming from minimax theory, infsup-convexity, is the adequate one for this kind of results.
We prove a local version of Gowers Ramsey-type theorem [21], as well as loc
The purpose of this paper is devoted to studying representation of measures of non generalized compactness, in particular, measures of noncompactness, of non-weak compactness, and of non-super weak compactness, etc, defined on Banach spaces and its applications. With the aid of a three-time order preserving embedding theorem, we show that for every Banach space $X$, there exist a Banach function space $C(K)$ for some compact Hausdorff space $K$, and an order-preserving affine mapping $mathbb T$ from the super space $mathscr B$ of all nonempty bounded subsets of $X$ endowed with the Hausdorff metric to the positive cone $C(K)^+$ of $C(K)$ such that for every convex measure, in particular, regular measure, homogeneous measure, sublinear measure of non generalized compactness $mu$ on $X$, there is a convex function $digamma$ on the cone $V=mathbb T(mathscr B)$ which is Lipschitzian on each bounded set of $V$ such that [digamma(mathbb T(B))=mu(B),;;forall;Binmathscr B.] As its applications, we show a class of basic integral inequalities related to an initial-value problem in Banach spaces, and prove a solvability result of the initial-value problem, which is an extension of some classical results due to Goebel, Rzymowski, and Bana{s}.
In this paper, we give the definability of bilinear singular and fractional integral operators on Morrey-Banach space, as well as their commutators and we prove the boundedness of such operators on Morrey-Banach spaces. Moreover, the necessary condition for BMO via the bounedness of bilinear commutators on Morrey-Banach space is also given. As a application of our main results, we get the necessary conditions for BMO via the bounedness of bilinear integral operators on weighted Morrey space and Morrey space with variable exponents. Finally, we obtain the boundedness of bilinear C-Z operator on Morrey space with variable exponents.
We study the existence of zeroes of mappings defined in Banach spaces. We obtain, in particular, an extension of the well-known Bolzano-Poincare-Miranda theorem to infinite dimensional Banach spaces. We also establish a result regarding the existence of periodic solutions to differential equations posed in an arbitrary Banach space.
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