The compact weak topology on a Banach space

Many of the fundamental research problems in the geometry of normed linear spaces can be loosely phrased as: Given a Banach space X and a class of Banach spaces Y does X contain a subspace Y ∈ Y? As a Banach space X is determined by its unit ball B x ≡ { x ∈ X :‖ x ‖ ≤ 1 } the problem can be rephrased in terms of the geometry of convex sets: Can a given unit ball B x be sliced with a subspace to obtain a set in some given class of unit balls? A result of this type is the famous theorem of Dvoretzky (see also [L], [M6], [M4], [MS], [FLM]).

We prove that the additive group (E*, τ k (E)) of an $${\mathcal {L}_\infty}$$ -Banach space E, with the topology τ k (E) of uniform convergence on compact subsets of E, is topologically isomorphic to a subgroup of the unitary group of some Hilbert space (is unitarily representable). This is the same as proving that the topological group (E*, τ k (E)) is uniformly homeomorphic to a subset of $${\ell_2^\kappa}$$ for some κ. As an immediate consequence, preduals of commutative von Neumann algebras or duals of commutative C*-algebras are unitarily representable in the topology of uniform convergence on compact subsets. The unitary representability of free locally convex spaces (and thus of free Abelian topological groups) on compact spaces, follows as well. The above facts cannot be extended to noncommutative von Neumann algebras or general Schwartz spaces.

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

Let denote the space of functions holomorphic on an open Riemann surface R, where has the topology of uniform convergence on compact sets. In this note, we characterize the dual space *. The result is not new, for it is implicitly contained in the more general results of Tillmann [5] and, in case R is planar, in those of Köthe [4]. However, the paper of Gunning and Narasimhan [3], which appeared subsequently, allows us to give a short proof of this important result. Actually, our characterization is a natural one in terms of differentials, while the Köthe-Tillmann characterization is in terms of functions, but we show that these two characterizations are isomorphic. We end our paper by using our characterization to prove an interpolation result. The second author gratefully acknowledges a helpful discussion with Professor George Szekeres.

In this paper, a new class of Banach spaces, termed as Banach spaces with property (MB), will be introduced. It is stated that a space X has property (MB) if every V -subset of X* is an L-subset of X* . We describe those spaces which have property (MB) . Also, we show that if a Banach space X has property (MB) and Banach space Y does not contain , then every operator is completely continuous.

A Weak Grothendieck Compactness Principle for Infinite Dimensional Banach Spaces Kaitlin Bjorkman The goal of this thesis is to give an exposition of the following recent result of Freeman, Lennard, Odell, Turett and Randrianantoanina [3] A Banach space has the Schur property if and only if every weakly compact set is contained in the closed convex hull of a weakly null sequence. This result complements an old result of Grothendieck (now called the Grothendieck Compactness Principle) stating that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. We include many of the relevant definitions and preliminary results which are required in the proofs of both of these theorems. Dissertation Director: Dr. Kevin Beanland A Weak Grothendieck Compactness Principle for Infinite Dimensional Banach Spaces

Equivalence, linearization, and decomposition of holomorphic operator functions

Gateaux Differentiation of Functions on a Certain Class of Banach Spaces

In this work, we study Bishop–Phelps cones (briefly, BP cones) given by an equation in Banach spaces. Due to the special form, these cones enjoy interesting properties. We show that nontrivial BP cones have given by an equation form a ‘large family’ in some sense in any Banach space and they can be used to characterize the strict convexity of the space. We obtain conditions for a convex cone to be included in or to be itself a BP cone given by an equation. Furthermore, we give an affirmative answer to the open question of whether these cones may possess a nonempty interior in infinite-dimensional spaces and introduce Lorentz cones in some classical Banach spaces of sequences as illustrations. We also obtain explicit representations for all BP cones given by an equation in some classical Banach spaces. Finally, we present some short applications in optimal control and approximation.

For the duality mapping on a Banach space the relation between lower semi-continuity and upper semi-continuity properties is explored, upper semi-continuity is characterized in terms of slices of the ball and upper semi-continuity properties are related to geometrical properties which imply that the space is an Asplund space. The duality mapping is a natural set-valued mapping from the unit sphere of a normed linear space into subsets of its dual sphere, and which for an inner product space is the mapping associating an element of the unit sphere with the corresponding continuous linear functional given by the inner product. It is an example of a subdifferential mapping of a continuous convex function (in this case, the norm), which is in turn a special kind of maximal monotone mapping. Cudia [4, p. 298] showed that the duality mapping is always upper semi-continuous when the space has the norm and the dual space has the weak* topology, and Kenderov [10, p. 67] extended this to maximal monotone mappings. Bonsall, Cain, and Schneider [3] used the property to prove the connectedness of the numerical range of an operator on a normed linear space. Along with the activity which culminated in StegalΓs theorem [15] characterizing an Asplund space as one whose dual has the Radon-Nikodym property, there has been some interest in finding geometrical conditions sufficient for a space to be Asplund. A Banach space X is an Asplund space if every continuous convex function defined on an open convex subset of X is strongly differentiate on a dense Gδ subset of its domain. Ekeland and Lebourg [6, p. 204] have shown that a Banach space is Asplund if there exists a strongly differentiate real function on the space with bounded nonempty support, in particular, if the space can be equivalently renormed to have norm strongly differentiable on the unit sphere. Using StegalΓs theorem, a result of Diestel and Faires [5, p. 625] gives that a Banach space is Asplund if the space can be equivalently renormed to be very smooth, that is, to be smooth and to have the single valued duality mapping continuous when the space has the norm and the dual space has the weak topology. Recently Smith and Sullivan [13, Theorem 15] have exhibited a more general condition which is sufficient for a Banach space to be Asplund. We show that

For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G,\(\widehat{G})\) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (H) of (G) that are easier to handle and show that \(\mathscr{C} (H)\) and \(\mathscr{C} (G/H)\) are large and embed, as a poset, in \(\mathscr{C}(G,τ)\). Important special results are: (i) if \(K\) is a compact subgroup of a locally quasi-convex group \(G\), then \(\mathscr{C}(G)\) and \(\mathscr{C}(G/K)\) are quasi-isomorphic (3.15); (ii) if (D) is a discrete abelian group of infinite rank, then \(\mathscr{C}(D)\) is quasi-isomorphic to the poset \(\mathfrak{F}_D\) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group \(G \) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset \( \mathscr{C} (G) \) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group \(X\), the group of null sequences \(G=c_0(X)\) with the topology of uniform convergence is studied. We prove that \(\mathscr{C}(G)\) is quasi-isomorphic to \(\mathscr{P}(\mathbb{R})\) (6.9).

It is shown that in a Grothendieck space with the Dunford-Pettis property, the class of well-bounded operators of type (B) coincides with the class of scalar-type spectral operators with real spectrum. It turns out that in such Banach spaces, analogues of the classical theorems of Hille-Sz. Nagy and Stone concerned with the integral representation of C0-semigroups of normal operators and strongly continuous unitary groups in Hilbert spaces, respectively, are of a very special nature.

Let X be a partially ordered real Banach space, a,b ∈ X with a ≤ b. Let φ be a bounded linear functional on X. We call X a Ben-Israel-Charnes space (or a B-C space, for short) if the linear program: Maximize 〈φ,x〉 subject to a ≤ x ≤ b has an optimal solution. Such problems have been shown to be important in solving a class of problems known as Interval Linear Programs. B-C spaces were introduced by the first author in his doctoral dissertation. In this paper we identify new classes of Banach spaces that are B-C spaces. We also present sufficient conditions under which answers are in the affirmative for the following questions: When is a closed subspace of a B-C space, a B-C space? Is the range of a bounded linear map from a Banach space into a B-C space, a B-C space? AMS Subject Classification: Primary: 90C48, 90C05; Secondary: 47N10.

Minimal cusco maps have applications in functional analysis, in optimization, in the study of weak Asplund spaces, in the study of differentiability of functions, etc. It is important to know their topological properties. Let X be a Hausdorff topological space, MC(X) be the space of minimal cusco maps with values in R and ?UC be the topology of uniform convergence on compacta. We study completemetrizability and cardinal invariants of (MC(X), ?UC). We prove that for two nondiscrete locally compact second countable spaces X and Y, (MC(X), ?UC) and (MC(Y), ?UC) are homeomorphic and they are homeomorphic to the space C(Ic) of continuous real-valued functions on Ic with the topology of uniform convergence.

For $p\geq 2$, let $E$ be a 2-uniformly smooth and $p$-uniformly convex real Banach space and let $A:E\rightarrow E^{\ast }$ be a Lipschitz and strongly monotone mapping such that $A^{-1}(0)\neq \emptyset$. For given $x_{1}\in E$, let $\{x_{n}\}$ be generated by the algorithm $x_{n+1}=J^{-1}(Jx_{n}-\unicode[STIX]{x1D706}Ax_{n})$, $n\geq 1$, where $J$ is the normalized duality mapping from $E$ into $E^{\ast }$ and $\unicode[STIX]{x1D706}$ is a positive real number in $(0,1)$ satisfying suitable conditions. Then it is proved that $\{x_{n}\}$ converges strongly to the unique point $x^{\ast }\in A^{-1}(0)$. Furthermore, our theorems provide an affirmative answer to the Chidume et al. open problem [‘Krasnoselskii-type algorithm for zeros of strongly monotone Lipschitz maps in classical Banach spaces’, SpringerPlus4 (2015), 297]. Finally, applications to convex minimization problems are given.