On summability domains

1973 ◽  
Vol 73 (2) ◽  
pp. 327-338 ◽  
Author(s):  
N. J. Kalton

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.

1973 ◽  
Vol 25 (3) ◽  
pp. 511-524 ◽  
Author(s):  
G. Bennett ◽  
N. J. Kalton

A sequence space is a vector subspace of the space ω of all real (or complex) sequences. A sequence space E with a locally convex topology τ is called a K- space if the inclusion map E → ω is continuous, when ω is endowed with the product topology . A K-space E with a Frechet (i.e., complete, metrizable and locally convex) topology is called an FK-space; if the topology is a Banach topology, then E is called a BK-space.


Author(s):  
D. J. H. Garling

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.


1989 ◽  
Vol 40 (1) ◽  
pp. 123-128 ◽  
Author(s):  
Charles Swartz

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.


1988 ◽  
Vol 37 (3) ◽  
pp. 383-388 ◽  
Author(s):  
W.J. Robertson ◽  
S.A. Saxon ◽  
A.P. Robertson

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.


Author(s):  
Camillo Trapani

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.


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