scholarly journals The Fréchet space $\omega $ admits a strictly stronger separable and quasicomplete locally convex topology

1981 ◽  
Vol 82 (4) ◽  
pp. 655-655
Author(s):  
Susanne Dierolf
Keyword(s):  
Fréchet Space ◽  
Frechet Space ◽  
Convex Topology ◽  
Locally Convex Topology ◽  
Locally Convex

Local analytic structure in certain commutative topological algebras

1972 ◽  
Vol 6 (2) ◽  
pp. 161-167 ◽  
Author(s):  
R.J. Loy
Keyword(s):  
Power Series ◽  
Topological Algebra ◽  
Fréchet Space ◽  
Analytic Structure ◽  
Frechet Space ◽  
Topological Algebras ◽  
Space Topology ◽  
Formal Power ◽  
Locally Convex Topology ◽  
Locally Convex

Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

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.

Interpolation in Separable Frechet Spaces with Applications to Spaces of Analytic Functions

1975 ◽  
Vol 27 (5) ◽  
pp. 1110-1113 ◽  
Author(s):  
Paul M. Gauthier ◽  
Lee A. Rubel
Keyword(s):  
Sufficient Conditions ◽  
Fréchet Space ◽  
Complex Numbers ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Linear Functional ◽  
Interpolating Sequence ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Spaces Of Analytic Functions

Let E be a separable Fréchet space, and let E* be its topological dual space. We recall that a Fréchet space is, by definition, a complete metrizable locally convex topological vector space. A sequence {Ln} of continuous linear functional is said to be interpolating if for every sequence {An} of complex numbers, there exists an ƒ ∈ E such that Ln(ƒ) = An for n = 1, 2, 3, … . In this paper, we give necessary and sufficient conditions that {Ln} be an interpolating sequence. They are different from the conditions in [2] and don't seem to be easily interderivable with them.

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.

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