On Isomorphisms of Locally Convex Spaces With Similar Biorthogonal Systems

1973 ◽  
Vol 16 (2) ◽  
pp. 179-183
Author(s):  
F. Bozel ◽  
T. Husain

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].

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
Zhi-Ang Zhou

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.


1967 ◽  
Vol 15 (4) ◽  
pp. 295-296 ◽  
Author(s):  
Sunday O. Iyahen

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.


1993 ◽  
Vol 48 (2) ◽  
pp. 209-249 ◽  
Author(s):  
Vladimir G. Pestov

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.


1975 ◽  
Vol 20 (4) ◽  
pp. 468-482 ◽  
Author(s):  
J. R. Giles ◽  
G. Joseph ◽  
D. O. Koehler ◽  
B. Sims

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.


1968 ◽  
Vol 9 (2) ◽  
pp. 111-118 ◽  
Author(s):  
S. O. Iyahen

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.


1993 ◽  
Vol 48 (1) ◽  
pp. 1-6
Author(s):  
J.C. Ferrando ◽  
L.M. Sánchez Ruiz

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.


1977 ◽  
Vol 20 (4) ◽  
pp. 317-327
Author(s):  
J. O. Popoola ◽  
I. Tweddle

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.


1973 ◽  
Vol 18 (3) ◽  
pp. 167-172 ◽  
Author(s):  
J. H. Webb

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).


Author(s):  
Thomas E. Gilsdorf

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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