An example of the barrelled space associated to C(X;E)

1981 ◽  
pp. 561-571 ◽  
Author(s):  
Jean Schmets
Keyword(s):  
Barrelled Space

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

scholarly journals Strictly barrelled disks in inductive limits of quasi-(LB)-spaces

1996 ◽  
Vol 19 (4) ◽  
pp. 727-732
Author(s):  
Carlos Bosch ◽  
Thomas E. Gilsdorf
Keyword(s):  
Convex Space ◽  
Inductive Limit ◽  
Linear Span ◽  
Locally Convex Space ◽  
Minkowski Functional ◽  
Closed Graph ◽  
Inductive Limits ◽  
Locally Convex ◽  
Barrelled Space

A strictly barrelled diskBin a Hausdorff locally convex spaceEis a disk such that the linear span ofBwith the topology of the Minkowski functional ofBis a strictly barrelled space. Valdivia's closed graph theorems are used to show that closed strictly barrelled disk in a quasi-(LB)-space is bounded. It is shown that a locally strictly barrelled quasi-(LB)-space is locally complete. Also, we show that a regular inductive limit of quasi-(LB)-spaces is locally complete if and only if each closed bounded disk is a strictly barrelled disk in one of the constituents.

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 Barrelled spaces and dense vector subspaces

1988 ◽  
Vol 37 (3) ◽  
pp. 383-388 ◽  
Author(s):  
W.J. Robertson ◽  
S.A. Saxon ◽  
A.P. Robertson
Keyword(s):  
Simple Proof ◽  
Structure Theorem ◽  
Vector Subspace ◽  
Hamel Basis ◽  
Convex Topology ◽  
Barrelled Spaces ◽  
Locally Convex Topology ◽  
Locally Convex ◽  
Barrelled Space ◽  
Vector Subspaces

This note presents a structure theorem for locally convex barrelled spaces. It is shown that, corresponding to any Hamel basis, there is a natural splitting of a barrelled space into a topological sum of two vector subspaces, one with its strongest locally convex topology. This yields a simple proof that a barrelled space has a dense infinite-codimensional vector subspace, provided that it does not have its strongest locally convex topology. Some further results and examples discuss the size of the codimension of a dense vector subspace.

scholarly journals The Completion of a Pseudo-Barrelled Space Is Pseudo-Barrelled

1995 ◽  
Vol 192 (2) ◽  
pp. 667-668
Author(s):  
S. Oluyemi
Keyword(s):  
Barrelled Space

scholarly journals On the stability of barrelled topologies, III

1980 ◽  
Vol 22 (1) ◽  
pp. 99-112 ◽  
Author(s):  
W.J. Robertson ◽  
I. Tweddle ◽  
F.E. Yeomans
Keyword(s):  
Sufficient Conditions ◽  
Vector Subspace ◽  
Finite Dimensional ◽  
Countable Dimension ◽  
The Stability ◽  
Bounded Sets ◽  
Barrelled Space

Let E be a barrelled space with dual F ≠ E*. It is shown that F has uncountable codimension in E*. If M is a vector subspace of E* of countable dimension with M ∩ F = {o}, the topology τ(E, F+M) is called a countable enlargement of τ(E, F). The results of the two previous papers are extended: it is proved that a non-barrelled countable enlargement always exists, and sufficient conditions for the existence of a barrelled countable enlargement are established, to include cases where the bounded sets may all be finite dimensional. An example of this case is given, derived from Amemiya and Kōmura; some specific and general classes of spaces containing a dense barrelled vector subspace of codimension greater than or equal to c are discussed.

scholarly journals A barrelled space without a basis

1970 ◽  
Vol 26 (3) ◽  
pp. 465-465 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Barrelled Space

scholarly journals The fit and flat components of barrelled spaces

1995 ◽  
Vol 51 (3) ◽  
pp. 521-528 ◽  
Author(s):  
Stephen A. Saxon ◽  
Ian Tweddle
Keyword(s):  
Continuum Hypothesis ◽  
Dense Subspace ◽  
Splitting Theorem ◽  
Hamel Basis ◽  
Barrelled Spaces ◽  
Barrelled Space ◽  
Complementary Subspaces ◽  
The Continuum

The Splitting Theorem says that any given Hamel basis for a (Hausdorff) barrelled space E determines topologically complementary subspaces Ec and ED, and that Ec is flat, that is, contains no proper dense subspace. By use of this device it was shown that if E is non-flat it must contain a dense subspace of codimension at least ℵ0; here we maximally increase the estimate to ℵ1 under the assumption that the dominating cardinal ∂ equals ℵ1 [strictly weaker than the Continuum Hypothesis (CH)]. A related assumption strictly weaker than the Generalised CH allows us to prove that ED is fit, that is, contains a dense subspace whose codimension in ED is dim (ED), the largest imaginable. Thus the two components are extreme opposites, and E itself is fit if and only if dim (ED) ≥ dim (Ec), in which case there is a choice of basis for which ED = E. Morover, E is non-flat (if and) only if the codimension of E′ is at least in E*. These results ensure latitude in the search for certain subspaces of E* transverse to E′, as in the barrelled countable enlargement (BCE) problem, and show that every non-flat GM-space has a BCE.

scholarly journals On a Barrelled Space of Class $\aleph_0$ and Measure Theory.

1992 ◽  
Vol 71 ◽  
pp. 96 ◽  
Author(s):  
J. C. Ferrando ◽  
L. M. Sánchez Ruiz
Keyword(s):  
Measure Theory ◽  
Barrelled Space

scholarly journals On the stability of barrelled topologies, I

1979 ◽  
Vol 20 (3) ◽  
pp. 385-395 ◽  
Author(s):  
W.J. Robertson ◽  
F.E. Yeomans
Keyword(s):  
Dual Space ◽  
Vector Spaces ◽  
Mackey Topology ◽  
Topological Vector Spaces ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Countable Dimension ◽  
The Stability ◽  
Locally Convex ◽  
Barrelled Space

This note investigates, for locally convex topological vector spaces, the question of how far the property of being barrelled is stable under small increase in the size of the dual space. If the dual F of a barrelled space E is enlarged by a finite dimensional vector space M, then E remains barrelled under the new Mackey topology τ(E, F+M). We discuss what happens when M has countable dimension.

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