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 Baire-Type Properties in Metrizable c0(Ω, X)

Axioms ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 6
Author(s):  
Salvador López-Alfonso ◽  
Manuel López-Pellicer ◽  
Santiago Moll-López
Keyword(s):  
Normed Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Metrizable Space ◽  
Complex Numbers ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Open Problems ◽  
Locally Convex

Ferrando and Lüdkovsky proved that for a non-empty set Ω and a normed space X, the normed space c0(Ω,X) is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered Baire-like. When X is a metrizable locally convex space, with an increasing sequence of semi-norms .n∈N defining its topology, then c0(Ω,X) is the metrizable locally convex space over the field K (of the real or complex numbers) of all functions f:Ω→X such that for each ε>0 and n∈N the set ω∈Ω:f(ω)n>ε is finite or empty, with the topology defined by the semi-norms fn=supf(ω)n:ω∈Ω, n∈N. Kąkol, López-Pellicer and Moll-López also proved that the metrizable space c0(Ω,X) is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p if and only if X is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p. The main result of this paper is that the metrizable c0(Ω,X) is baireled if and only if X is baireled, and its proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented.

scholarly journals Stability and closed graph theorems in classes of bornological spaces

1982 ◽  
Vol 23 (2) ◽  
pp. 151-162
Author(s):  
T. K. Mukherjee ◽  
W. H. Summers
Keyword(s):  
Inductive Limit ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Closed Graph ◽  
Quotient Spaces ◽  
Direct Sum ◽  
Bornological Space ◽  
Normed Vector ◽  
Locally Convex

In the general theory of locally convex spaces, the idea of inductive limit is pervasive, with quotient spaces and the less obvious notion of direct sum being among the instances. Bornological spaces provide another important example. As is well known (cf. [7]), a Hausdorff locally convex space E is bornological if, and only if, E is an inductive limit of normed vector spaces. Going even further in this direction, a complete Hausdorff bornological space is an inductive limit of Banach spaces.

(LF)-spaces with absolute bases

1970 ◽  
Vol 67 (2) ◽  
pp. 283-286 ◽  
Author(s):  
G. Bennett ◽  
J. B. Cooper
Keyword(s):  
Complex Plane ◽  
Real Line ◽  
Convex Space ◽  
Inductive Limit ◽  
Locally Convex Space ◽  
The Real ◽  
Continuous Seminorm ◽  
Image Position ◽  
Locally Convex ◽  
Absolute Basis

Suppose E is a locally convex space over a field K which can be the real line or the complex plane. Then a basis for E is a sequence (xk) of elements of E such that, if x ∈ E, x can be expressed uniquely aswhere ξk ∈K for each k. If this representation converges absolutely, i.e. iffor every continuous seminorm p on E, then (xk) is called an absolute basis for E. If the mappings x → ξk from E into K are continuous for each k, then (xk) is a Schauder basis for E. The purpose of this paper is to prove some results for (LF)-spaces with bases and to use them to extend some theorems due to Pietsch. We recall that an (F)-space is a complete metrizable locally convex space and an (LF)-space the inductive limit of a strictly increasing sequence of (F)-spaces (En, τn) such that τn+1|En = τn for all n.

scholarly journals ON QUASI *-BARRELLED SPACES

1990 ◽  
Vol 21 (4) ◽  
pp. 341-344
Author(s):  
S. G. GAYAL
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
New Class ◽  
Barrelled Spaces ◽  
Locally Convex ◽  
Barrelled Space ◽  
Convex Spaces

In this paper, a new class of .ocally convex spaces, called quasi *- barrelled spaces is introduced. These spaces are characterized by : A locally convex space $E$ is Quasi *-barrelled if every bornivorous *-barrel in $E$ is a neighbourhood of $O$ in $E$. This class of spaces is a generalization of quasi-barrelled spaces and *-barrelled spaces (K.Anjaneyulu; Gayal : Jour. Math. Phy. Sci. Madras, 1984). Some properties of quasi *-barrelled spaces are sturued. Lastly one example each of (i) a quasi *-barrelled space which is not quasi-barrelled. (ii) a quasi *-barrelled space which is not *-barrelled. is given.

Closed graph theorems for generalized inductive limit topologies

1977 ◽  
Vol 82 (1) ◽  
pp. 67-83 ◽  
Author(s):  
W. Ruess
Keyword(s):  
Inductive Limit ◽  
Locally Convex Space ◽  
Subsequent Paper ◽  
Closed Graph ◽  
Strict Topology ◽  
Locally Compact ◽  
Special Cases ◽  
Bounded Sets ◽  
Locally Convex Topology ◽  
Locally Convex

SummaryThe object of this and a subsequent paper is to investigate the locally convex structure of several strict topologies that are generalizations of R. C. Buck's strict topology β on C(S), S locally compact Hausdorff. If the topology τ of a locally convex space (lcs) (X, τ) is any of these strict topologies, then it is localizable on every absorbing disc T in X, i.e. it is the finest locally convex topology on X agreeing with τ on T. Topologies of this kind are said to be (L)-topologies. As our main tools for the analysis of the structure of strict topologies, we deduce in this paper several closed graph theorems for spaces of type (L). In particular, it is shown that every semi-Montel lcs with a fundamental sequence of bounded sets and every Bτ-complete Schwartz space belongs to the class Bτ(L) of all lcs Y with the property that every closed linear map from any (L)-space X into Y is continuous. Further closed graph theorems are established and many of the known closed graph theorems are deduced as special cases of our results. Moreover, the problem of Bτ-completeness of locally convex spaces belonging to Bτ(L) is considered.

scholarly journals A maximum principle

1979 ◽  
Vol 28 (1) ◽  
pp. 23-26
Author(s):  
Kung-Fu Ng
Keyword(s):  
Maximum Principle ◽  
Extreme Point ◽  
Convex Space ◽  
Maximal Element ◽  
Locally Convex Space ◽  
Compact Set ◽  
Natural Manner ◽  
Upper Semicontinuous ◽  
Nonempty Compact ◽  
Locally Convex

AbstractLet K be a nonempty compact set in a Hausdorff locally convex space, and F a nonempty family of upper semicontinuous convex-like functions from K into [–∞, ∞). K is partially ordered by F in a natural manner. It is shown among other things that each isotone, upper semicontinuous and convex-like function g: K → [ – ∞, ∞) attains its K-maximum at some extreme point of K which is also a maximal element of K.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 46 A 40.

scholarly journals In what spaces is every closed normal cone regular?

1970 ◽  
Vol 17 (2) ◽  
pp. 121-125 ◽  
Author(s):  
C. W. McArthur
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Unconditional Basis ◽  
Normal Cone ◽  
Closed Subspace ◽  
Locally Convex Space ◽  
Regular Part ◽  
Fréchet Space ◽  
Sequential Completeness ◽  
Locally Convex

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

scholarly journals Extending Edgar's ordering to locally convex spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 175-188
Author(s):  
Neill Robertson
Keyword(s):  
Convex Space ◽  
Dual Pair ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex ◽  
Mackey Topologies

By the term “locally convex space”, we mean a locally convex Hausdorff topological vector space (see [17]). We shall denote the algebraic dual of a locally convex space E by E*, and its topological dual by E′. It is convenient to think of the elements of E as being linear functionals on E′, so that E can be identified with a subspace of E′*. The adjoint of a continuous linear map T:E→F will be denoted by T′:F′→E′. If 〈E, F〈 is a dual pair of vector spaces, then we shall denote the corresponding weak, strong and Mackey topologies on E by α(E, F), β(E, F) and μ(E, F) respectively.

scholarly journals A Fenchel-Rockafellar type duality theorem for maximization

1979 ◽  
Vol 20 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Convex Space ◽  
Duality Theorem ◽  
Locally Convex Space ◽  
Bounded Subset ◽  
Convex Functional ◽  
Lower Semi ◽  
Locally Convex

We prove that sup(f-h)(E) = sup(h*-f*)(E*), where f is a proper lower semi-continuous convex functional on a real locally convex space E, h: E → = [-∞, +∞] is an arbitrary-functional and, f*, h* are their convex conjugates respectively. When h = δG, the indicator of a bounded subset G of E, this yields a formula for sup f(G).

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