Baire-Type Properties in Metrizable c0(Ω, X)

Ferrando and Lüdkovsky proved that for a non-empty set Ω and a normed space X, the normed space c0(Ω,X) is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered Baire-like. When X is a metrizable locally convex space, with an increasing sequence of semi-norms .n∈N defining its topology, then c0(Ω,X) is the metrizable locally convex space over the field K (of the real or complex numbers) of all functions f:Ω→X such that for each ε>0 and n∈N the set ω∈Ω:f(ω)n>ε is finite or empty, with the topology defined by the semi-norms fn=supf(ω)n:ω∈Ω, n∈N. Kąkol, López-Pellicer and Moll-López also proved that the metrizable space c0(Ω,X) is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p if and only if X is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p. The main result of this paper is that the metrizable c0(Ω,X) is baireled if and only if X is baireled, and its proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented.

Distinguished Property in Tensor Products and Weak* Dual Spaces

A local convex space E is said to be distinguished if its strong dual Eβ′ has the topology β(E′,(Eβ′)′), i.e., if Eβ′ is barrelled. The distinguished property of the local convex space CpX of real-valued functions on a Tychonoff space X, equipped with the pointwise topology on X, has recently aroused great interest among analysts and Cp-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space CpX is distinguished if and only if any function f∈RX belongs to the pointwise closure of a pointwise bounded set in CX. The extensively studied distinguished properties in the injective tensor products CpX⊗εE and in Cp(X,E) contrasts with the few distinguished properties of injective tensor products related to the dual space LpX of CpX endowed with the weak* topology, as well as to the weak* dual of Cp(X,E). To partially fill this gap, some distinguished properties in the injective tensor product space LpX⊗εE are presented and a characterization of the distinguished property of the weak* dual of Cp(X,E) for wide classes of spaces X and E is provided.

$$\omega ^\omega $$-Base and infinite-dimensional compact sets in locally convex spaces

A locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ { U α : α ∈ ω ω } at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ U β ⊆ U α for all $$\alpha \le \beta $$ α ≤ β . The class of lcs with an $$\omega ^{\omega }$$ ω ω -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ D ′ ( Ω ) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ φ endowed with the finest locally convex topology has an $$\omega ^\omega $$ ω ω -base but contains no infinite-dimensional compact subsets. It turns out that $$\varphi $$ φ is a unique infinite-dimensional locally convex space which is a $$k_{\mathbb {R}}$$ k R -space containing no infinite-dimensional compact subsets. Applications to spaces $$C_{p}(X)$$ C p ( X ) are provided.

L-concave derived internal relations and L-convex derived enclosed relations

Derived operator plays an important role in describing algebraic properties of many mathematical structures such as topology, matroid, convergence and convex space. In this paper, we present the notions of L-concave derived internal relation space and L-convex derived enclosed relation space by which we characterize L-concave space and L-convex space. Based on this, we further introduce some other structures such as L-concave derived hull space and L-convex derived hull space. We find that these spaces are isomorphic to L-concave space and L-convex space.

Subordination for sequentially equicontinuous equibounded $$C_0$$-semigroups

We consider operators A on a sequentially complete Hausdorff locally convex space X such that $$-A$$ - A generates a (sequentially) equicontinuous equibounded $$C_0$$ C 0 -semigroup. For every Bernstein function f we show that $$-f(A)$$ - f ( A ) generates a semigroup which is of the same ‘kind’ as the one generated by $$-A$$ - A . As a special case we obtain that fractional powers $$-A^{\alpha }$$ - A α , where $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) , are generators.

The arity of convex spaces

The arity of convex spaces is a numerical feature which shows the ability of finite subsets spanning to the whole space via the hull operators. This paper gives it a formal and strict definition by introducing the truncation of convex spaces. The relations that between the arity of quotient spaces and the original spaces, that between the arity of subspaces and superspaces, that between the arity of product spaces and factors spaces, and that between the arity of disjoint sums and term spaces, are systematically studied. A mistake of a formula in [M. Van De Vel, Theory of Convex Structures, North-Holland, Amsterdam, 1993] is corrected. It is shown that a convex space is Alexandrov iff its arity is 1. The convex structures with arity ≤n are equivalent to structured sets with n-restricted hull operators.

L-interval spaces and some of their properties

In this paper, notions of L-interval spaces and L-2-arity convex spaces are introduced. It is showed that there is a Galois’s connection between the category of L-convex spaces and the category of L-interval spaces. In particular, the category of L-2-arity convex spaces can be embedded in the category of L-interval spaces as a coreflective subcategory. Further, some properties of L-interval spaces are introduced including L-geometric (resp. L-Peano, L-Pasch and L-sand-glass) property. It is proved that an L-2-arity convex space is an L-JHC convex space iff its segment operator has L-Peano property. It is also proved that an L-JHC convex space with an L-idempotent segment operator has L-sand-glass property. Further, it is also proved that an L-idempotent interval space having L-Peano+L-Pasch property has L-geometric property and L-sand-glass property.

Improving KKM Theory on General Metric Type Spaces

A generalized metric type space is a generic name for various spaces similar to hyperconvex metric spaces or extensions of them. The purpose of this article is to introduce some KKM theoretic works on generalized metric type spaces and to show that they can be improved according to our abstract convex space theory. Most of these works are chosen on the basis that they can be improved by following our theory. Actually, we introduce abstracts of each work or some contents, and add some comments showing how to improve them.