On the finest locally convex topology agreeing with a given topology on a sequence of absolutely convex sets

1972 ◽  
Vol 198 (1) ◽  
pp. 57-80 ◽  
Author(s):  
W. Roelcke
Keyword(s):  
Convex Sets ◽  
Convex Topology ◽  
Locally Convex Topology ◽  
Locally Convex

The β- and γ-duality of sequence spaces

1967 ◽  
Vol 63 (4) ◽  
pp. 963-981 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Linear Space ◽  
Sequence Spaces ◽  
Topological Properties ◽  
Product Topology ◽  
Convex Topology ◽  
Locally Convex Topology ◽  
Locally Convex

A K-space (E, τ) is a linear space E of sequences with a locally convex topology τ for which the inclusion map: (E, τ) → (ω, product topology) is continuous. In (2) topological properties of K-spaces were determined directly from properties of the space E and the topology τ. It is, however, very natural to consider duality properties of K-spaces and the purpose of this paper is to determine some of these properties.

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.

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 UnboundedC*-seminorms, biweights, and*-representations of partial*-algebras: A review

2006 ◽  
Vol 2006 ◽  
pp. 1-34 ◽  
Author(s):  
Camillo Trapani
Keyword(s):  
Representation Theory ◽  
State Of The Art ◽  
Complete Analysis ◽  
Sesquilinear Forms ◽  
Convex Topology ◽  
Relevant Role ◽  
Partial Algebras ◽  
The Given ◽  
Locally Convex Topology ◽  
Locally Convex

The notion of (unbounded)C*-seminorms plays a relevant role in the representation theory of*-algebras and partial*-algebras. A rather complete analysis of the case of*-algebras has given rise to a series of interesting concepts like that of semifiniteC*-seminorm and spectralC*-seminorm that give information on the properties of*-representations of the given*-algebraAand also on the structure of the*-algebra itself, in particular whenAis endowed with a locally convex topology. Some of these results extend to partial*-algebras too. The state of the art on this topic is reviewed in this paper, where the possibility of constructing unboundedC*-seminorms from certain families of positive sesquilinear forms, called biweights, on a (partial)*-algebraAis also discussed.

The order-bound topology

1972 ◽  
Vol 71 (2) ◽  
pp. 321-327 ◽  
Author(s):  
Yau-Chuen Wong
Keyword(s):  
Vector Space ◽  
Positive Cone ◽  
Order Topology ◽  
Bounded Subset ◽  
Partially Ordered ◽  
Convex Topology ◽  
Ordered Vector Space ◽  
Partially Ordered Vector Space ◽  
Locally Convex Topology ◽  
Locally Convex

Let (E, C) be a partially ordered vector space with positive cone C. The order-bound topology Pb(6) (order topology in the terminology of Schaefer(9)) on E is the finest locally convex topology for which every order-bounded subset of E is topologically bounded.

On summability domains

1973 ◽  
Vol 73 (2) ◽  
pp. 327-338 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Convex Topology ◽  
Bk Space ◽  
Complex Sequences ◽  
Image Position ◽  
Locally Convex Topology ◽  
Locally Convex

We denote by ω the space of all complex sequences with the topology given by the semi-normswhere δn(x) = xn. An FK-space, E, is a subspace of ω on which there exists a complete metrizable locally convex topology τ, such that the inclusion (E, τ) ⊂ ω is continuous; if τ is given by a single norm then E is a BK-space.

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