Locally convex topologies and the convex compactness property

1974 ◽  
Vol 75 (1) ◽  
pp. 45-50 ◽  
Author(s):  
E. G. Ostling ◽  
A. Wilansky
Keyword(s):  
Compact Set ◽  
Compactness Property ◽  
Original Topology ◽  
Krein’S Theorem ◽  
Locally Convex Topologies ◽  
The Subject ◽  
Convex Compactness ◽  
Convex Closure ◽  
Bounded Sets ◽  
Locally Convex

1. Introduction. A locally convex space is said to have the convex compactness property (sometimes abbreviated to cc) if the absolutely convex closure of each compact set is compact. This important property is the subject of Krein's theorem (3) 24.5(4′). It is strictly weaker than bounded completeness and can sometimes be substituted for that assumption; for example, a useful result, related to the Banach–Mackey theorem, says that in a space with cc, all admissible topologies have the same bounded sets (5). As another example, it is well known that if X is bornological, X′ is strongly complete (see (1), theoreml); but if X has cc as well, we can strengthen this result to conclude, (2) 19C, that X′ is complete with its Mackey topology, indeed with the topology Ta (using the notation of section 3), where T is the original topology of X.

scholarly journals Banach-Dieudonné Theorem Revisited

2003 ◽  
Vol 75 (1) ◽  
pp. 69-83 ◽  
Author(s):  
Montserrat Bruguera ◽  
Elena Martín-Peinador
Keyword(s):  
Topological Group ◽  
Locally Convex Spaces ◽  
Compact Open Topology ◽  
Compactness Property ◽  
Character Group ◽  
Abelian Topological Group ◽  
Continuous Convergence ◽  
Convergence Structure ◽  
Convex Compactness ◽  
Locally Convex

AbstractWe prove that in the character group of an abelian topological group, the topology associated (in a standard way) to the continuous convergence structure is the finest of all those which induce the topology of simple convergence on the corresponding equicontinuous subsets. If the starting group is furthermore metrizable (or even almost metrizable), we obtain that such a topology coincides with the compact-open topology. This result constitutes a generalization of the theorem of Banach-Dieudonné, which is well known in the theory of locally convex spaces.We also characterize completeness, in the class of locally quasi-convex metrizable groups, by means of a property which we have called the quasi-convex compactness property, or briefly qcp (Section 3).

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.

Compactly determined locally convex topologies

1972 ◽  
Vol 196 (2) ◽  
pp. 91-100 ◽  
Author(s):  
Horacio Porta
Keyword(s):  
Locally Convex Topologies ◽  
Locally Convex

scholarly journals A generalisation of Mackey's theorem and the uniform boundedness principle

1989 ◽  
Vol 40 (1) ◽  
pp. 123-128 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Uniform Boundedness ◽  
Mackey Topology ◽  
Convex Topology ◽  
Uniform Boundedness Principle ◽  
Bounded Sets ◽  
Locally Convex Topology ◽  
Locally Convex

We construct a locally convex topology which is stronger than the Mackey topology but still has the same bounded sets as the Mackey topology. We use this topology to give a locally convex version of the Uniform Bouudedness Principle which is valid without any completeness or barrelledness assumptions.

The Dual of Semigroup Algebras with Certain Locally Convex Topologies

2006 ◽  
Vol 73 (3) ◽  
pp. 367-376 ◽  
Author(s):  
S. Maghsoudi ◽  
R. Nasr-Isfahani ◽  
A. Rejali
Keyword(s):  
Semigroup Algebras ◽  
Locally Convex Topologies ◽  
Locally Convex

scholarly journals Order-bounded convergence structures on spaces of continuous functions

1979 ◽  
Vol 28 (1) ◽  
pp. 39-61 ◽  
Author(s):  
M. Schroder
Keyword(s):  
Vector Lattice ◽  
Continuous Functions ◽  
Subject Classification ◽  
Linear Functionals ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions ◽  
Locally Convex Topologies ◽  
Locally Convex ◽  
Compact Convergence

AbstractThis paper deals with solid topologies and convergence structures on the vector-lattice CX (the set of all continuous real-valued functions on a space X): the closed ideals and locally convex topologies associated with such structures are studied in particular. The work stems from E. Hewitt's paper on bounded linear functionals, touches on the classical theorems of L. Nachbin, T. Shirota and others (determining when the topology of compact convergence is barrelled or bornological), and extends some recent results on the duality between x and CX.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 54 C 35; secondary 54 A 20.

Lines and Hyperplanes associated with Families of Closed and Bounded Sets in Conjugate Banach Spaces

1970 ◽  
Vol 22 (5) ◽  
pp. 933-938
Author(s):  
M. Edelstein
Keyword(s):  
General Setting ◽  
Locally Convex Space ◽  
Finite Family ◽  
Topological Spaces ◽  
Compact Sets ◽  
Straight Line ◽  
Straight Lines ◽  
Bounded Sets ◽  
Infinite Set ◽  
Locally Convex

Let be a family of sets in a linear space X. A hyperplane π is called a k-secant of if π intersects exactly k members of . The existence of k-secants for families of compact sets in linear topological spaces has been discussed in a number of recent papers (cf. [3–7]). For X normed (and a finite family of two or more disjoint non-empty compact sets) it was proved [5] that if the union of all members of is an infinite set which is not contained in any straight line of X, then has a 2-secant. This result and related ones concerning intersections of members of by straight lines have since been extended in [4] to the more general setting of a Hausdorff locally convex space.

scholarly journals NOTE ABOUT LINDELÖF Σ-SPACES υX

2011 ◽  
Vol 85 (1) ◽  
pp. 114-120
Author(s):  
J. KA̧KOL ◽  
M. LÓPEZ-PELLICER
Keyword(s):  
Functional Analysis ◽  
Convex Space ◽  
Locally Convex Space ◽  
Countable Tightness ◽  
Pointwise Topology ◽  
Bounded Sets ◽  
General Fact ◽  
Locally Convex

AbstractThe paper deals with the following problem: characterize Tichonov spaces X whose realcompactification υX is a Lindelöf Σ-space. There are many situations (both in topology and functional analysis) where Lindelöf Σ (even K-analytic) spaces υX appear. For example, if E is a locally convex space in the class 𝔊 in sense of Cascales and Orihuela (𝔊 includes among others (LM ) -spaces and (DF ) -spaces), then υ(E′,σ(E′,E)) is K-analytic and E is web-bounded. This provides a general fact (due to Cascales–Kakol–Saxon): if E∈𝔊, then σ(E′,E) is K-analytic if and only if σ(E′,E) is Lindelöf. We prove a corresponding result for spaces Cp (X) of continuous real-valued maps on X endowed with the pointwise topology: υX is a Lindelöf Σ-space if and only if X is strongly web-bounding if and only if Cp (X) is web-bounded. Hence the weak* dual of Cp (X) is a Lindelöf Σ-space if and only if Cp (X) is web-bounded and has countable tightness. Applications are provided. For example, every E∈𝔊 is covered by a family {Aα :α∈Ω} of bounded sets for some nonempty set Ω⊂ℕℕ.

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