scholarly journals On Homomorphisms, Open Operators and their Adjoints

2001 ◽  
Vol 8 (4) ◽  
pp. 823-844
Author(s):  
D. Zarnadze
Keyword(s):  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Convex Spaces ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Convex Spaces

Abstract The well-known A. Grothendieck's theorem on a homomorphism between locally convex spaces is generalized to the case of topologies which are incompatible with dualities. On the basis of this theorem, necessary and sufficient conditions are obtained for a weak homomorphism (resp. its adjoint operator, resp. its double adjoint operator) to be again a homomorphism in various topologies of the initial (resp. dual, resp. bidual) spaces. Some new classes of pairs of locally convex spaces satisfying these conditions are established. The results obtained have enabled us to reveal new properties of frequently encountered homomorphisms and weakly open operators, as well as to strengthen and generalize some well-known results.

scholarly journals Necessary and sufficient conditions for precompact sets to be metrisable

2006 ◽  
Vol 74 (1) ◽  
pp. 7-13 ◽  
Author(s):  
J.C. Ferrando ◽  
J. Kasakol ◽  
M. López Pellicer
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Convex Spaces ◽  
Compact Sets ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Convex Spaces ◽  
Compact Subsets

This self-contained paper characterises those locally convex spaces whose (weakly) precompact (respectively, compact) subsets are metrisable. Applications and examples are provided. Our approach also applies to get Cascales-Orihuela's, Valdivia's and Robertson's metrisation theorems for (pre)compact sets.

Projections and embeddings of locally convex operator spaces and their duals

1984 ◽  
Vol 96 (2) ◽  
pp. 321-323 ◽  
Author(s):  
Jan H. Fourie ◽  
William H. Ruckle
Keyword(s):  
Dual Space ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Operator Spaces ◽  
Complemented Subspace ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Continuous Linear Operators

AbstractLet E, F be Hausdorff locally convex spaces. In this note we consider conditions on E and F such that the dual space of the space Kb (E, F) (of quasi-compact operators) is a complemented subspace of the dual space of Lb (E, F) (of continuous linear operators). We obtain necessary and sufficient conditions for Lb(E, F) to be semi-reflexive.

scholarly journals Global Schauder decompositions of locally convex spaces

2007 ◽  
Vol 101 (1) ◽  
pp. 65
Author(s):  
Milena Venkova
Keyword(s):  
Convex Space ◽  
Entire Functions ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Convex Spaces

We define global Schauder decompositions of locally convex spaces and prove a necessary and sufficient condition for two spaces with global Schauder decompositions to be isomorphic. These results are applied to spaces of entire functions on a locally convex space.

XI.—On Unconditional Convergence in Topological Vector Spaces

1969 ◽  
Vol 68 (2) ◽  
pp. 145-157 ◽  
Author(s):  
A. P. Robertson
Keyword(s):  
Sufficient Conditions ◽  
Direct Proof ◽  
Vector Spaces ◽  
Unconditional Convergence ◽  
Partial Sums ◽  
Locally Convex Spaces ◽  
Topological Groups ◽  
Topological Vector Spaces ◽  
Necessary And Sufficient ◽  
Locally Convex

SynopsisFor a series of elements of a topological vector space, necessary and sufficient conditions are found, in terms of the set of finite partial sums, for unconditional convergence and for the corresponding Cauchy condition. The extent to which these results remain valid for topological groups is investigated. A new and direct proof, for locally convex spaces, is given of the theorem of Orlicz.

scholarly journals Regularization of cylindrical processes in locally convex spaces

2020 ◽  
pp. 2050027
Author(s):  
Christian A. Fonseca-Mora
Keyword(s):  
Convex Space ◽  
Lévy Process ◽  
Sufficient Conditions ◽  
Locally Convex Space ◽  
Levy Process ◽  
Locally Convex Spaces ◽  
Locally Convex ◽  
Strong Dual ◽  
Convex Spaces

Let [Formula: see text] be a locally convex space and let [Formula: see text] denote its strong dual. In this paper, we introduce sufficient conditions for the existence of a continuous or a càdlàg [Formula: see text]-valued version to a cylindrical process defined on [Formula: see text]. Our result generalizes many other known results on the literature and their different connections will be discussed. As an application, we provide sufficient conditions for the existence of a [Formula: see text]-valued càdlàg Lévy process version to a given cylindrical Lévy process in [Formula: see text].

scholarly journals Completion in the bidual

1968 ◽  
Vol 8 (2) ◽  
pp. 238-241 ◽  
Author(s):  
R. Nielsen
Keyword(s):  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Locally Convex

Let E, Ê, and E′ denote a locally convex linear Hausdorff space, completion of E and the dual of E, respectively. It has been observed that Ê is a subspace of E″ under certain conditions on E. It is the primary goal of this paper to give necessary and sufficient conditions for the Ê ⊂ E″ to be valid. Such conditions are found and are given Theorem 4. With a variation of the technique used, several equivalent characterizations of semi-reflexive spaces are given in Theorem 5. The nationa throughtout will follow that in [2].

On ∧-Mackey convergence in locally convex spaces

1984 ◽  
Vol 96 (3) ◽  
pp. 495-500
Author(s):  
Jan H. Fourie
Keyword(s):  
Convergence Property ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Necessary And Sufficient Condition ◽  
Banach Sequence Space ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Banach Sequence ◽  
Convex Spaces

In this note we introduce the concepts of Λ-Mackey sequence, Λ-Mackey convergence property, Λ-Schwartz family and associated Λ-Schwartz family and consider some applications of these to locally convex spaces. Hereby Λ denotes a Banach sequence space with the AK-property — the results of this paper generalize those in [4] where the case Λ = I1 is considered. We obtain a dual characterization of those locally convex spaces which satisfy the Λ-Mackey convergence property and characterize the dual Λ-Schwartz spaces in terms of the SM-property which is introduced in [10]. Finally, necessary and sufficient condition for a locally convex space to be ultra-bornological is proved.

scholarly journals New Characterizations of k-Uniformly Extremely Convex Banach Spaces

2018 ◽  
Vol 2018 ◽  
pp. 1-4
Author(s):  
Suyalatu Wulede
Keyword(s):  
Banach Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Convex Spaces

Two characterizations of k-uniformly extremely convex spaces are given in this paper. As special cases for k=1, two necessary and sufficient conditions of uniform extreme convexity are presented and a characteristic inequality of uniformly extremely convex spaces is given.

An Inverse Mapping Theorem in Frechet Spaces

1986 ◽  
Vol 29 (2) ◽  
pp. 238-245
Author(s):  
Henri-François Gautrin ◽  
Khaldoun Imam ◽  
Tapio Klemola ◽  
Jean-Marc Terrier
Keyword(s):  
Sufficient Conditions ◽  
Locally Convex Spaces ◽  
Fréchet Spaces ◽  
Inverse Mapping ◽  
Inverse Mapping Theorem ◽  
Locally Convex ◽  
Convex Spaces

AbstractWithin the framework of a-differentiability, introduced by H. R. Fischer in locally convex spaces, sufficient conditions for an inverse mapping theorem between Fréchet spaces are established.

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