A Gelfand-Phillips Property with Respect to the Weak Topology

We consider a Gelfand-Phillips type property for the weak topology. The main results that we obtain are (1) for certain Banach spaces, E^?_? F inherits this property from E and F, and (2) the spaces L^p(μ, E) have this property when E does. A subset A of a Banach space E is a limited set if every (bounded linear) operator T:E → c₀ maps A onto a relatively compact subset of c₀. The Banach space E has the Gelfand-Phillips property if every limited set is relatively compact. In this note, we study the analogous notions set in the weak topology. Thus we say that A ? E is a Grothendieck set if every T: E → c₀ maps A onto a relatively weakly compact set; and E is said to have the weak type GP property if every Grothendieck set in E is relatively weakly compact. In the papers [3, 4 and 6], it is shown among other results that the ?-tensor product E and the spaces L^p(μ, E) inherit the Gelfand-Phillips property from E and F. In this paper, we study the same questions for the weak type GP property. It is easily verified that continuous linear images of Grothendieck sets are Grothendieck and that the weak type GP property is inherited by subspaces. Among the spaces with the weak type GP property one easily finds the separable spaces, and more generally, spaces with a weak* sequentially compact dual ball. Also, C(K) spaces where K is (DCSC) are weak type GP (see [3] and the discussion before Corollary 4 below). A Grothendieck space (a Banach space whose unit ball is a Grothendieck set) has the weak type GP if and only if it is reflexive.     

Banach Disks and Marrelledness Properties of Metrizalbe <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript> (Ω, <Emphasis Type="Italic">X</Emphasis>)

Let be a non-empty set and X a metrizable locally convex space. We show that the metrizable locally convex space c₀ (, X) is p-barrelled (totally barrelled) if and only if X is p-barrelled (totally barrelled). Some applications for closed graph theorems are included.     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous. (Received 24 March 1999; in final form 14 February 2000)     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous.     

On countable strict inductive limits

In this paper we prove that if E is the strict inductive limit of a sequence of Mackey spaces {E_n} such that for every positive integer n, the topological dual space of E_n, E′_n, provided with the Mackey topology μ(E′_n,E_n), is ultrabornological, then the topological dual space E′ of E, provided with the Mackey topology μ(E′,E), is ultrabornological. We also prove that if E is a strict (LF)-space and G a closed subspace of E′ [μ(E′,E)] such that E′[μ(E′,E)] /G is sequentially complete, then E′[μ(E′,E)]/G is complete.     

EMBEDDING C0 IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

Abstract

We show that the normed space of μ-measurable Pettis integrable functions on a probability space with values in a Banach space X contains a copy of the sequence space c₀ if and only if X contains a copy of c₀. In this case, if the probability μ has infinite range, a copy of c₀ consisting of μ-measurable functions can be found, such that it is complemented in the bigger space of all weakly μ-measurable Pettis integrable functions.     

Onc 0 sequences in Banach spaces

A Banach space has property (S) if every normalized weakly null sequence contains a subsequences equivalent to the unit vector basis ofc ₀. We show that the equivalence constant can be chosen “uniformly”, i.e., independent of the choice of the normalized weakly null sequence. Furthermore we show that a Banach space with property (S) has property (u). This solves in the negative the conjecture that a separable Banach space with property (u) not containingl ₁ has a separable dual. This is part of this author's Ph.D. dissertation prepared at The University of Texas at Austin under the supervision of H. P. Rosenthal.     

On tree characterizations of <Emphasis Type="Italic">G</Emphasis><Subscript>δ</Subscript>-embeddings and some Banach spaces

We show that a one-to-one bounded linear operator T from a separable Banach space E to a Banach space X is a G _δ-embedding if and only if every T-null tree in S _E has a branch which is a boundedly complete basic sequence. We then consider the notions of regulators and skipped blocking decompositions of Banach spaces and show, in a fairly general set up, that the existence of a regulator is equivalent to that of special skipped blocking decomposition. As applications, the following results are obtained. (a) A separable Banach space E has separable dual if and only if every w*-null tree in S _E * has a branch which is a boundedly complete basic sequence. (b) A Banach space E with separable dual has the point of continuity property if and only if every w-null tree in S _E has a branch which is a boundedly complete basic sequence. We also give examples to show that the tree hypothesis in both the cases above cannot be replaced in general with the assumption that every normalized w*-null (w-null in (b)) sequence has a subsequence which is a boundedly complete basic sequence. The research of S. Dutta was supported in part by the Institute for Advanced Studies in Mathematics at Ben-Gurion University of the Negev. The research of V. P. Fonf was supported in part by Israel Science Foundation, Grant No. 139/03.     

Some isomorphically polyhedral orlicz sequence spaces

A Banach space is polyhedral if the unit ball of each of its finite dimensional subspaces is a polyhedron. It is known that a polyhedral Banach space has a separable dual and isc ₀-saturated, i.e., each closed infinite dimensional subspace contains an isomorph ofc ₀. In this paper, we show that the Orlicz sequence spaceh _M is isomorphic to a polyhedral Banach space if lim _t→0 M(Kt)/M(t)=∞ for someK<∞. We also construct an Orlicz sequence spaceh _M which isc ₀-saturated, but which is not isomorphic to any polyhedral Banach space. This shows that beingc ₀-saturated and having a separable dual are not sufficient for a Banach space to be isomorphic to a polyhedral Banach space.     

Cylindrical multidimensional surfaces in Lobachevsky space

We study complete, strongly parabolic metrics with constant relative index of nullity _c=k and complete, strongly parabolic surfaces with constant index of relative nullityv=k in a constant curvature space Rⁿ(c) under the assumption that there exists a surface orthogonal to the fibers of total geodesity, and if c<0, under the additional condition that the orthogonal surface be totally umbilical. For c>0 we have _c=l;v=l, for c=0 the Riemannian manifold is a metric product of the metric of R^l–k and the Euclidean space E^k while the surface is a cylinder in Euclidean space. For c<0 the metric has a special form and the surface is a cylindrical surface in Lobachevsky space.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 18–27, 1990.     

On the Density of Positive Proper Efficient Points in a Normed Space

In the context of vector optimization and generalizing cones with bounded bases, we introduce and study quasi-Bishop-Phelps cones in a normed space X. A dual concept is also presented for the dual space X*. Given a convex subset A of a normed space X partially ordered by a closed convex cone S with a base, we show that, if A is weakly compact, then positive proper efficient points are sequentially weak dense in the set E(A, S) of efficient points of A; in particular, the connotation weak dense in the above can be replaced by the connotation norm dense if S is a quasi-Bishop-Phelps cone. Dually, for a convex subset of X* partially ordered by the dual cone S ⁺, we establish some density results of positive weak* efficient elements of A in E(A, S ⁺).     

On the Nonseparable Subspaces of J(η) and C([1, η])

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 𝓁₂(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c₀(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω₁) (resp. C([1, ω₁])) contains a complemented copy of 𝓁₂(ω₁) (resp.c₀(ω₁)). As consequence, we give examples (J(ω₁), C([1, ω₁]), C(V), V being the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i.e., for every subspace Y ⊆ X we have that Dens (Y) = w*–Dens (Y*)), in spite of these spaces are not weakly Lindelof determined (WLD).     

Weak selections and weak orderability of function spaces

It is proved that for a zero-dimensional space X, the function space C _p (X, 2) has a Vietoris continuous selection for its hyperspace of at most 2-point sets if and only if X is separable. This provides the complete affirmative solution to a question posed by Tamariz-Mascarúa. It is also obtained that for a strongly zero-dimensional metrizable space E, the function space C _p (X, E) is weakly orderable if and only if its hyperspace of at most 2-point sets has a Vietoris continuous selection. This provides a partial positive answer to a question posed by van Mill and Wattel.     

Chain conditions and weak topologies

We study conditions on Banach spaces close to separability. We say that a topological space is pcc if every point-finite family of open subsets of the space is countable. For a Banach space E, we say that E is weakly pcc if E, equipped with the weak topology, is pcc, and we also consider a weaker property: we say that E is half-pcc if every point-finite family consisting of half-spaces of E is countable. We show that E is half-pcc if, and only if, every bounded linear map E→c₀(ω₁) has separable range. We exhibit a variety of mild conditions which imply separability of a half-pcc Banach space. For a Banach space C(K), we also consider the pcc-property of the topology of pointwise convergence, and we note that the space Cp(K) may be pcc even when C(K) fails to be weakly pcc. We note that this does not happen when K is scattered, and we provide the following example:

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There exists a non-metrizable scattered compact Hausdorff space K with C(K) weakly pcc.

     

The weak McShane integral

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a σ-finite outer regular quasi Radon measure space (S,Σ, T, µ) into a Banach space X and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function f from S into X is weakly McShane integrable on each measurable subset of S if and only if it is Pettis and weakly McShane integrable on S. On the other hand, we prove that if an X-valued function is weakly McShane integrable on S, then it is Pettis integrable on each member of an increasing sequence (S _l ) _l?1 of measurable sets of finite measure with union S. For weakly sequentially complete spaces or for spaces that do not contain a copy of c ₀, a weakly McShane integrable function on S is always Pettis integrable. A class of functions that are weakly McShane integrable on S but not Pettis integrable is included.     

Coding into Ramsey sets

In [6] W. T. Gowers formulated and proved a Ramsey-type result which lies at the heart of his famous dichotomy for Banach spaces. He defines the notion of weakly Ramsey set of block sequences of an infinite dimensional Banach space and shows that every analytic set of block sequences is weakly Ramsey. We show here that Gowers’ result follows quite directly from the fact that all G_δ sets are weakly Ramsey, if the Banach space does not contain c₀, and from the fact that all F_σδ sets are weakly Ramsey, in the case of an arbitrary Banach space. We also show that every result obtained by the application of Gowers’ theorem to an analytic set can also be obtained by applying the Theorem to a F_σδ set (or to a G_δ set if the space does not contain c₀). This fact explains why the only known applications of this technique are based on very low-ranked Borel sets (open, closed, F_σ, or G_δ).     

Intersections of Fréchet Schwartz spaces and their duals

In this article the structure of the intersections of a Fréchet Schwartz space F and a (DFS)-space E=ind _n E _n is investigated. A complete characterization of the locally convex properties of E ⋃ F is given. This space is boraological if and only if the inductive limit E + F is complete. The results are based on recent progress on the structure of (LF)-spaces. The article includes examples of (FS)-spaces F and (DFS)-spaces E such that there are sequentially continuous linear forms on E ⋃ F which are not continuous, thus answering a question of Langenbruch. Acknowledgement: The results in this article were obtained during the author’s stay at the University of Paderborn, Germany, during the academic year 1994/95. The support of the Alexander von Humboldt Stiftung is greatly appreciated. The content of the article was presented as an invited paper in a Special Session of the AMS meeting in New York in April, 1996.     

On normal and compact operators on Banach spaces

For some normal operators (T=H+iK) on a Banach spaceX we study the dual space of the Banach algebraA (H, K) assuming thatX* is weakly complete and we study the decompositionX=Ker (T) ⊕ (TX)⁻ for spacesX ⊅c ₀.     

Basic Sequences in the Dual of a Fréchet Space

In this paper we study some properties of basic sequences in the dual of a Fréechet space. As a consequence we obtain that if E is a Fréechet space with the property that for each closed subspace F of E and each bounded subset B of E/F there is a bounded subset A of E with φ(A) = B, where φ denotes the canonical surjection of E onto E/F, then one of the following conditions is at least satisfied: 1. E is a Banach space, 2. E is a Schwartz space, 3. E is the product of a Banach space by ω. Finally, we also obtain some results concerning totally reflexive spaces.     

Sufficient conditions for existence of extrema in linear programming

It is shown that, on a closed convex subset X of a real Hausdorff locally convex space E, a continuous linear functional x′ on E has an extremum at an extreme point of X, provided X contains no line and X ∩ (x′)^?1 (λ₀) is non-empty and weakly compact for some real λ₀. It is also shown that any weakly locally compact closed convex subset of E that contains no line is the sum of its asymptotic cone and the closed convex hull of its extreme points.