scholarly journals The Mackey problem for free locally convex spaces

2018 ◽  
Vol 30 (6) ◽  
pp. 1339-1344 ◽  
Author(s):  
Saak Gabriyelyan
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Free Locally Convex Space ◽  
Locally Convex ◽  
Convex Spaces

Abstract It is known that the free locally convex space {L(X)} on a space X is metrizable only if X is finite and that {L(X)} is barrelled if and only if X is discrete. We significantly generalize these results by proving that {L(X)} is a Mackey space if and only if X is discrete. Noting that real locally convex spaces which are Mackey groups are always Mackey spaces, but that the converse is false, it is also proved here that {L(X)} is a Mackey group if and only if it is a Mackey space.

Free Locally Convex Spaces and the k-space Property

2014 ◽  
Vol 57 (4) ◽  
pp. 803-809 ◽  
Author(s):  
S. S. Gabriyelyan
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Discrete Space ◽  
Locally Convex Spaces ◽  
Tychonoff Space ◽  
Free Locally Convex Space ◽  
Locally Convex ◽  
Space Property ◽  
Convex Spaces

AbstractLet L(X) be the free locally convex space over a Tychonoff space X. Then L(X) is a k-space if and only if X is a countable discrete space. We prove also that L(D) has uncountable tightness for every uncountable discrete space D.

scholarly journals The H∞-optimization in locally convex spaces

2002 ◽  
Vol 15 (2) ◽  
pp. 91-103
Author(s):  
Chuan-Gan Hu ◽  
Li-Xin Ma
Keyword(s):  
Control Theory ◽  
Convex Space ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Model Matching ◽  
Matching Error ◽  
Locally Convex ◽  
Convex Spaces

In this paper, the ordinary H∞-control theory is extended to locally convex spaces through the form of a parameter. The algorithms of computing the infimal model-matching error and the infimal controller are presented in a locally convex space. Two examples with the form of a parameter are enumerated for computing the infimal model-matching error and the infimal controller.

scholarly journals Some remarks on countably barrelled and countably quasibarrelled spaces

1967 ◽  
Vol 15 (4) ◽  
pp. 295-296 ◽  
Author(s):  
Sunday O. Iyahen
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Countable Union ◽  
Bounded Subset ◽  
Linear Subspaces ◽  
Barrelled Spaces ◽  
Locally Convex ◽  
Barrelled Space ◽  
Convex Spaces

Barrelled and quasibarrelled spaces form important classes of locally convex spaces. In (2), Husain considered a number of less restrictive notions, including infinitely barrelled spaces (these are the same as barrelled spaces), countably barrelled spaces and countably quasibarrelled spaces. A separated locally convex space E with dual E' is called countably barrelled (countably quasibarrelled) if every weakly bounded (strongly bounded) subset of E' which is the countable union of equicontinuous subsets of E' is itself equicontinuous. It is trivially true that every barrelled (quasibarrelled) space is countably barrelled (countably quasibarrelled) and a countably barrelled space is countably quasibarrelled. In this note we give examples which show that (i) a countably barrelled space need not be barrelled (or even quasibarrelled) and (ii) a countably quasibarrelled space need not be countably barrelled. A third example (iii)shows that the property of being countably barrelled (countably quasibarrelled) does not pass to closed linear subspaces.

scholarly journals Global Schauder decompositions of locally convex spaces

2007 ◽  
Vol 101 (1) ◽  
pp. 65
Author(s):  
Milena Venkova
Keyword(s):  
Convex Space ◽  
Entire Functions ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Convex Spaces

We define global Schauder decompositions of locally convex spaces and prove a necessary and sufficient condition for two spaces with global Schauder decompositions to be isomorphic. These results are applied to spaces of entire functions on a locally convex space.

Subspaces with property (b) in locally convex spaces of quasi-barrelled type

1980 ◽  
Vol 88 (2) ◽  
pp. 331-337 ◽  
Author(s):  
Bella Tsirulnikov
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Dense Subspace ◽  
Locally Convex Spaces ◽  
Linear Hull ◽  
Bounded Set ◽  
Property B ◽  
Locally Convex ◽  
Strong Dual ◽  
Convex Spaces

A subspace G of a locally convex space E has property (b) if for every bounded set B of E the codimension of G in the linear hull of G ∪ B is finite, (5). Extending the results of (5) and (14), we prove that, if the strong dual of E is complete, then subspaces with property (b) inherit the following properties of E: σ-evaluability, evaluability, the property of being Mazur, semibornological and bornological. We also prove that a dense subspace with property (b) of a Mazur space is sequentially dense, and of a semibornological space – dense in the sense of Mackey (locally dense, following M. Valdivia).

scholarly journals An Egerev's theorem for vector functions

1985 ◽  
Vol 31 (3) ◽  
pp. 451-462
Author(s):  
P. Jimenez Guerra ◽  
Jose L. de Maria Gonzalez
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Locally Convex ◽  
Convex Spaces

In this paper some results of Egorov's theorem type are given for functions with values in locally convex spaces and Riesz's theorem is proved for functions taking values in a sequentially complete locally convex space.

scholarly journals On a functional differential equation in locally convex spaces

1983 ◽  
Vol 27 (2) ◽  
pp. 269-283
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Differential Equation ◽  
Functional Differential Equation ◽  
Convex Space ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Functional Differential ◽  
Nonlinear Maps ◽  
Locally Convex ◽  
Convex Spaces

The notion of accretiveness for multi-valued nonlinear maps is defined in locally convex spaces and it is used to obtain a locally convex space version of a result of M.G. Crandall and J.A. Nohel.

scholarly journals Regularization of cylindrical processes in locally convex spaces

2020 ◽  
pp. 2050027
Author(s):  
Christian A. Fonseca-Mora
Keyword(s):  
Convex Space ◽  
Lévy Process ◽  
Sufficient Conditions ◽  
Locally Convex Space ◽  
Levy Process ◽  
Locally Convex Spaces ◽  
Locally Convex ◽  
Strong Dual ◽  
Convex Spaces

Let [Formula: see text] be a locally convex space and let [Formula: see text] denote its strong dual. In this paper, we introduce sufficient conditions for the existence of a continuous or a càdlàg [Formula: see text]-valued version to a cylindrical process defined on [Formula: see text]. Our result generalizes many other known results on the literature and their different connections will be discussed. As an application, we provide sufficient conditions for the existence of a [Formula: see text]-valued càdlàg Lévy process version to a given cylindrical Lévy process in [Formula: see text].

An invariant theorem in duality

2010 ◽  
Vol 47 (3) ◽  
pp. 299-310
Author(s):  
Yunyan Yang ◽  
Ronglu Li
Keyword(s):  
Convex Space ◽  
Weak Topology ◽  
Absolute Convergence ◽  
Convergent Series ◽  
Dual Pair ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Locally Convex ◽  
Convex Spaces

The concept of absolute convergence for series is generalized to locally convex spaces and an invariant theorem for absolutely convergent series in duality is established: when a locally convex space X is weakly sequentially complete, an admissible topology which is strictly stronger than the weak topology on X in the dual pair ( X,X′ ) is given such that it has the same absolutely convergent series as the weak topology in X .

Embedding locally convex spaces in nearly exotic spaces

1971 ◽  
Vol 70 (3) ◽  
pp. 399-400 ◽  
Author(s):  
N. T. Peck
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Locally Convex ◽  
Convex Spaces

AbstractIt is proved that every locally convex space is linearly homeomorphic to a subspace of a space which admits no non-trivial continuous linear functionals.

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