Strict inductive limits in locally convex cones

We propose a definition for strict inductive limits in locally convex cones. By this definition, we prove that the strict inductive limit of a sequence of locally convex cones with the strict separation property has the same strict separation property. Also we establish that the strict inductive limit of a sequences of separated cones is separated too. Finally we verify barreledness for this strict inductive limit.     

Locally convex quotient lattice cones

Walter Roth has investigated certain equivalence relations on locally convex cones in [W. Roth, Locally convex quotient cones, J. Convex Anal. 18, No. 4, 903–913 (2011)] which give rise to the definition of a locally convex quotient cone. In this paper, we investigate some special equivalence relations on a locally convex lattice cone by which the locally convex quotient cone becomes a lattice. In the case of a locally convex solid Riesz space, this reduces to the known concept of locally convex solid quotient Riesz space. We prove that the strict inductive limit of locally convex lattice cones is a locally convex lattice cone. We also study the concept of locally convex complete quotient lattice cones.     

Duality on locally convex cones

The theory of locally convex cones as a branch of functional analysis was presented by K. Keimel and W. Roth in [K. Keimel, W. Roth, Ordered Cones and Approximation, Lecture Notes in Math., vol. 1517, Springer-Verlag, Heidelberg, 1992]. We study some more results about dual cones and adjoint operators on locally convex cones. Moreover we introduce the concept of the uniformly precompact sets and discuss their relations with σ-bounded sets. Some results obtained about inductive limit, projective limit, metrizability and quotients of locally convex cones.     

Projective limits of weighted LB-spaces of holomorphic functions

Countable projective limits of countable inductive limits, called PLB-spaces, of weighted Banach spaces of continuous functions have recently been investigated by Agethen, Bierstedt and Bonet. We extend their investigation to the case of holomorphic functions regarding the same type of questions, i.e. we analyze locally convex properties in terms of the defining double sequence of weights and study the interchangeability of projective and inductive limit.     

Non-conflicting ordering cones and vector optimization in inductive limits

Let (E, ξ)= ind (E_n, ξ_n) be an inductive limit of a sequence (E_n, ξ_n)_{n∈ N} of locally convex spaces and let every step (E_n, ξ_n) be endowed with a partial order by a pointed convex (solid) cone S_n. In the framework of inductive limits of partially ordered locally convex spaces, the notions of lastingly efficient points, lastingly weakly efficient points and lastingly globally properly efficient points are introduced. For several ordering cones, the notion of non-conflict is introduced. Under the requirement that the sequence (S_n)_{n∈ N} of ordering cones is non-conflicting, an existence theorem on lastingly weakly efficient points is presented. From this, an existence theorem on lastingly globally properly efficient points is deduced.     

Sequentially Complete Inductive Limits and Regularity

A notion of an almost regular inductive limits is introduced. Every sequentially complete inductive limit of arbitrary locally convex spaces is almost regular.     

Some topologies on a Šerstnev probabilistic normed space

In this paper we will introduce two other topologies, coarser than the so-called strong topology, on a class of Šerstnev probabilistic normed spaces, and obtain some important properties of these topologies. We will show that under the first topology, denoted by τ₀, our probabilistic normed space is decomposable into the topological direct sum of a normable subspace and the subspace of probably null elements. Under the second topology, which is in fact the inductive limit topology of a family of locally convex topologies, the dual space becomes a locally convex topological vector space.     

Countable inductive limits of frechet algebras

We show that the Gelfand-Mazur theorem holds for countable inductive limits of Frechet algebras (we do not assume that the homomorphisms which define the inductive limit are continuous, or one-to-one). This question is motivated by the fact that the spectrum of some elements of such an algebra may be empty. We also discuss in detail a countable inductive limit of Frechet algebras of holomorphic functions, which provides an elementary, but seminal, counterexample to the biinvariant subspace problem for complete, reflexive, locally convex spaces.     

Reflexivity of Inductive Limits

An inductive locally convex limit of reflexive topological spaces is reflexive iff it is almost regular.     

Hypercyclic operators on quasi-Mazur spaces

Modifying the method of Ansari, we give some criteria for hypercyclicity of quasi-Mazur spaces. They can be applied to judging hypercyclicity of non-complete and non-metrizable locally convex spaces. For some special locally convex spaces, for example, Köthe (LF)-sequence spaces and countable inductive limits of quasi-Mazur spaces, we investigate their hypercyclicity. As we see, bounded biorthogonal systems play an important role in the construction of Ansari. Moreover, we obtain characteristic conditions respectively for locally convex spaces having bounded sequences with dense linear spans and for locally convex spaces having bounded absorbing sets, which are useful in judging the existence of bounded biorthogonal systems.     

Characterization of Solid Cones

The author gives a dual characterization of solid cones in locally convex spaces. From this the author obtains some criteria for judging convex cones to be solid in various kinds of locally convex spaces. Using a general expression of the interior of a solid cone, the author obtains a number of necessary and sufficient conditions for convex cones to be solid in the framework of Banach spaces. In particular, the author gives a dual relationship between solid cones and generalized sharp cones. The related known results are improved and extended.     

On Solidness of Polar Cones

We investigate the properties of cones whose polars are solid in different polar topologies. By a standard duality argument, we obtain a number of necessary and sufficient conditions for closed convex cones to be solid in various locally convex spaces. From this, we can deduce easily the extensions of previous related results. Furthermore, we construct a class of closed convex cones in some Banach spaces, which are not solid but whose polars satisfy the angle property. This solves the Han conjecture in the negative.     

拟平移不变拓扑锥与局部β-凸空间的共轭锥

[1]中提出的局部β-凸分析问题从本质上来说是一种非线性凸分析问题 .为了刻画和研究局部β-凸空间 X的共轭锥 X*β ,本文在抽象凸锥上引进具有拟平移不变性质的拓扑结构 ,第一部分重点研究局部生成拓扑锥与赋范拓扑锥 .第二部分将这两种拓扑锥的一般理论应用于局部 β - 凸空间的共轭锥 X*β 的研究 ,得到 (X*β,U| A)与 (X*β ,‖‖ )的局部生成性与完备性定理等 .     

P-Adic Semi-Montel Spaces and Polar Inductive Limits

In this paper we give a characterization of p-adic semi-Montel spaces which allows us to describe the finest polar semi-Montel topology coarser than the original topology of a locally convex space. As an application we derive that every polar semi-Montel space is a polar inductive limit of a family of nuclear spaces. We also pay attention to the connection with compactifying operators.     

Boundedly Compatible Webs and Strict Mackey Convergence

We look for characterizations of those locally convex spaces that satisfy the strict Mackey convergence condition within the context of spaces with webs. We will say that a locally convex space has a boundedly compatible web if it has a web of absolutely convex sets whose members behave like zero neighborhoods in a metrizable locally convex space. It will be shown that these locally convex spaces satisfy the strict Mackey convergence condition. One consequence of this result will be a characterization of boundedly retractive inductive limits. We will also prove that if E is locally complete and webbed, then the strict Mackey convergence condition is equivalent to E having a boundedly compatible web.     

A uniform boundedness theorem for locally convex cones

We prove a uniform boundedness theorem for families of linear operators on ordered cones. Using the concept of locally convex cones we introduce the notions of barreled cones and of weak cone-completeness. Our main result, though no straightforward generalization of the classical case, implies the Uniform Boundedness Theorem for Fréchet spaces.

     

Some Questions of Heinrich on Ultrapowers of Locally Convex Spaces

In this note we treat some open problems of Heinrich on ultrapowers of locally convex spaces. In section 1 we investigate the localization of bounded sets in the full ultrapower of a locally convex space, in particular the coincidence of the full and the bounded ultrapower, mainly concentrating in the case of (DF)-spaces. In section 2 we provide a partial answer to a question of Heinrich on commutativity of strict inductive limits and ultrapowers. In section 3 we analyze the relation between some natural candidates for the notion of superreflexivity in the setting of Fréchet spaces. We give an example of a Fréchet-Schwartz space which is not the projective limit of a sequence of superreflexive Banach spaces.     

Some new results on ε-Tensor products of locally convex spaces

The paper contains the proofs of three new propositions on ε-Tensor products of locally convex spaces. The first two of these propositions are on the ε-tensor product of inductive limits of locally convex spaces. The third proposition is on integral bilinear forms. For inductive tensor products and π-tensor products, some results on properties of permanence appear in A. Grothendieck’s famous thesis. We prove, in the present paper, some properties of permanence of ε-tensor products.     

New existence results for efficient points in locally convex spaces ordered by supernormal cones

In this paper, the definition of supernormality for convex cones in locally convex spaces is discussed in detail on many interesting examples. Starting from the new direction for the study of the existence of efficient points (Pareto type optimums) in locally convex spaces offered by the concept of supernormal (nuclear) cone, we establish some existence results for the efficient points using boundedness and completeness of conical sections induced by non-empty subsets and we specify properties for the sets of efficient points beside important remarks     

Linear Operators Generated by a Countable Number of Quasi-differential Expressions

The theory of linear ordinary quasi-differential operators has been considered in Lebesgue locally integrable spaces on a single interval of the real line. Such spaces are not Banach spaces but can be considered as complete, locally convex, linear topological spaces where the topology is derived from a countable family of semi-norms. The first conjugate space can also be defined as a complete, locally convex, linear topological space but now with the topology derived as a strict inductive limit. This article extends the previous single interval results to the case when a finite or countable number of intervals of the real line is considered. Conjugate and preconjugate linear quasi-differential operators are defined and relationships between these operators are developed.