On Isomorphisms of Locally Convex Spaces With Similar Biorthogonal Systems

1973 ◽  
Vol 16 (2) ◽  
pp. 179-183
Author(s):  
F. Bozel ◽  
T. Husain
Keyword(s):  
Locally Convex Spaces ◽  
Linear Spaces ◽  
Biorthogonal Systems ◽  
Prove Theorem ◽  
Barrelled Spaces ◽  
The Relationship ◽  
Locally Convex ◽  
Convex Spaces

The relationship between bases and isomorphisms (i.e. linear homeomorphisms) between complete metrizable linear spaces has been studied with great interest by Arsove and Edwards (see [1] and [2]). We prove (Theorem 1) that in the case of B-complete barrelled spaces, similar generalized bases imply existence of an isomorphism. This result was also proved by Dyer and Johnson [4], so we do not give a proof. We show (Theorem 6) that if one assumes that the bases are Schauder and similar, then Theorem 1 holds for countably barrelled spaces. We use Theorem 1 to advantage (Theorems 2-5) to show that one can improve some results due to Davis [3].

scholarly journals The Relationship between Two Kinds of Generalized Convex Set-Valued Maps in Real Ordered Linear Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Zhi-Ang Zhou
Keyword(s):  
Convex Set ◽  
Locally Convex Spaces ◽  
Linear Spaces ◽  
Algebraic Interior ◽  
Vector Closure ◽  
Ordered Linear Spaces ◽  
The Relationship ◽  
Locally Convex ◽  
Convex Spaces

A new notion of the ic-cone convexlike set-valued map characterized by the algebraic interior and the vector closure is introduced in real ordered linear spaces. The relationship between the ic-cone convexlike set-valued map and the nearly cone subconvexlike set-valued map is established. The results in this paper generalize some known results in the literature from locally convex spaces to linear spaces.

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 Universal arrows to forgetful functors from categories of topological algebra

1993 ◽  
Vol 48 (2) ◽  
pp. 209-249 ◽  
Author(s):  
Vladimir G. Pestov
Keyword(s):  
Functional Analysis ◽  
Lie Algebras ◽  
Topological Algebra ◽  
Locally Convex Spaces ◽  
Topological Groups ◽  
Open Problems ◽  
Relationship Of ◽  
The Relationship ◽  
Locally Convex ◽  
Convex Spaces

We survey the present trends in theory of universal arrows to forgetful functors from various categories of topological algebra and functional analysis to categories of topology and topological algebra. Among them are free topological groups, free locally convex spaces, free Banach-Lie algebras, and more. An accent is put on the relationship of those constructions with other areas of mathematics and their possible applications. A number of open problems is discussed; some of them belong to universal arrow theory, and other may become amenable to the methods of this theory.

scholarly journals On numerical ranges of operators on locally convex spaces

1975 ◽  
Vol 20 (4) ◽  
pp. 468-482 ◽  
Author(s):  
J. R. Giles ◽  
G. Joseph ◽  
D. O. Koehler ◽  
B. Sims
Keyword(s):  
Numerical Range ◽  
Linear Operators ◽  
Locally Convex Spaces ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Range Theory ◽  
Locally Convex ◽  
Convex Spaces

Numerical range theory for linear operators on normed linear spaces and for elements of normed algebras is now firmly established and the main results of this study are conveniently presented by Bonsall and Duncan in (1971) and (1973). An extension of the spatial numerical range for a class of operators on locally convex spaces was outlined by Moore in (1969) and (1969a), and an extension of the algebra numerical range for elements of locally m-convex algebras was presented by Giles and Koehler (1973). It is our aim in this paper to contribute further to Moore's work by extending the concept of spatial numerical range to a wider class of operators on locally convex spaces.

scholarly journals Semiconvex spaces

1968 ◽  
Vol 9 (2) ◽  
pp. 111-118 ◽  
Author(s):  
S. O. Iyahen
Keyword(s):  
General Linear ◽  
Topological Spaces ◽  
Locally Convex Spaces ◽  
Parallel Study ◽  
Barrelled Spaces ◽  
Locally Convex ◽  
Convex Spaces

Many of the techniques and notions used to study various important theorems in locally convex spaces are not effective for general linear topological spaces. In [4], a study is made of notionsin general linear topological spaces which can be used to replace barrelled, bornological, and quasi-barrelled spaces. The present paper contains a parallel study in the context of semiconvex spaces.

scholarly journals Countably quasi-suprabarrelled spaces

1993 ◽  
Vol 48 (1) ◽  
pp. 1-6
Author(s):  
J.C. Ferrando ◽  
L.M. Sánchez Ruiz
Keyword(s):  
Locally Convex Spaces ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Barrelled Spaces ◽  
Permanence Properties ◽  
Locally Convex ◽  
Convex Spaces

In this paper we obtain some permanence properties of a class of locally convex spaces located between quasi-suprabarrelled spaces and quasi-totally barrelled spaces, for which a closed graph theorem is given.

scholarly journals A variant of separability in dual systems

1977 ◽  
Vol 20 (4) ◽  
pp. 317-327
Author(s):  
J. O. Popoola ◽  
I. Tweddle
Keyword(s):  
Locally Convex Spaces ◽  
Closed Graph ◽  
Dual Systems ◽  
Barrelled Spaces ◽  
Locally Convex ◽  
Convex Spaces

In (12) we introduced the concept of essential separability and used it to define two classes of locally convex spaces, δ-barrelled spaces and infra-δ-spaces, which serve as domain and range spaces respectively in certain closed graph theorems (12, Theorems 3 and 7). In this note we continue the study of these ideas. The relevant definitions are reproduced below.

scholarly journals Countable-codimensional subspaces of locally convex spaces

1973 ◽  
Vol 18 (3) ◽  
pp. 167-172 ◽  
Author(s):  
J. H. Webb
Keyword(s):  
Hausdorff Space ◽  
Locally Convex Spaces ◽  
Countable Intersection ◽  
Barrelled Spaces ◽  
Locally Convex ◽  
Convex Spaces

A barrel in a locally convex Hausdorff space E[τ] is a closed absolutely convex absorbent set. A σ-barrel is a barrel which is expressible as a countable intersection of closed absolutely convex neighbourhoods. A space is said to be barrelled (countably barrelled) if every barrel (σ-barrel) is a neighbourhood, and quasi-barrelled (countably quasi-barrelled) if every bornivorous barrel (σ-barrel) is a neighbourhood. The study of countably barrelled and countably quasi-barrelled spaces was initiated by Husain (2).

scholarly journals Mackey convergence and quasi-sequentially webbed spaces

1991 ◽  
Vol 14 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Thomas E. Gilsdorf
Keyword(s):  
Related Problem ◽  
Partial Solution ◽  
Convergence Condition ◽  
Fast Convergence ◽  
Locally Convex Spaces ◽  
Barrelled Spaces ◽  
Partial Description ◽  
Locally Convex ◽  
Convex Spaces

The problem of characterizing those locally convex spaces satisfying the Mackey convergence condition is still open. Recently in [4], a partial description was given using compatible webs. In this paper, those results are extended by using quasi-sequentially webbed spaces (see Definition 1). In particular, it is shown that strictly barrelled spaces satisfy the Mackey convergence condition and that they are properly contained in the set of quasi-sequentially webbed spaces. A related problem is that of characterizing those locally convex spaces satisfying the so-called fast convergence condition. A partial solution to this problem is obtained. Several examples are given.

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