scholarly journals Bounded sets in fast complete inductive limits

1984 ◽  
Vol 7 (3) ◽  
pp. 615-617 ◽  
Author(s):  
Jan Kucera ◽  
Carlos Bosch
Keyword(s):  
Locally Convex Spaces ◽  
Inductive Limits ◽  
Bounded Sets ◽  
Locally Convex ◽  
Convex Spaces

LetE1⊂E2⊂…be a sequence of locally convex spaces with all identity maps:En→En+1continuous andE=indlim Enfast complete. Then each set bounded inEis also bounded in someEniff for any Banach diskBbounded inEandn∈N, the closure ofB⋂EninBis bounded in someEm. This holds, in particular, if all spacesEnare webbed.

scholarly journals Characterizations of collectively precompact and semi-precompact opterators on topological vector spaces

1979 ◽  
Vol 28 (2) ◽  
pp. 179-188 ◽  
Author(s):  
M. V. Deshpande ◽  
S. M. Padhye
Keyword(s):  
Subject Classification ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Totally Bounded ◽  
Primary 47 ◽  
Bounded Sets ◽  
Locally Convex ◽  
Convex Spaces

AbstractCharacterizations of collectively precompact and collectively semi-precompact sets of operators on topological vector spaces are obtained. These lead to the characterization of totally bounded sets of semi-precompact operators on locally convex spaces.1980 Mathematics subject classification (Amer. Math. Soc): primary 47 B 05, 47 D 15; secondary 46 A 05, 46 A 15.

scholarly journals On two new classes of locally convex spaces

1983 ◽  
Vol 28 (3) ◽  
pp. 383-392 ◽  
Author(s):  
Kazuaki Kitahara
Keyword(s):  
Locally Convex Spaces ◽  
Inductive Limits ◽  
Permanence Properties ◽  
Montel Space ◽  
Definition Of ◽  
Locally Convex ◽  
Convex Spaces

The purpose of this paper is to introduce two new classes of locally convex spaces which contain the classes of semi-Montel and Montel spaces. Further we give some examples and study some properties of these classes. As to permanence properties, these classes have similar properties to semi-Montel and Montel spaces except strict inductive limits and these classes are not always preserved under their completions. We shall call these two classes as β–semi–Montel and β–Montel spaces. A β–semi–Montel space is obtained by replacing the word “bounded” by “strongly bounded” in the definition of a semi–Montel space. If a β–semi–Montel space is infra-barrelled, we call the space β–Montel.

scholarly journals Bounded sets and dual strong sequences in locally convex spaces

2006 ◽  
Vol 13 (2) ◽  
pp. 295-304
Author(s):  
Bella Tsirulnikov
Keyword(s):  
Locally Convex Spaces ◽  
Bounded Sets ◽  
Locally Convex ◽  
Convex Spaces

On weak compactness of bounded sets in Banach and locally convex spaces

2000 ◽  
Vol 52 (6) ◽  
pp. 837-846 ◽  
Author(s):  
P. I. Kogut ◽  
V. S. Mel'nik
Keyword(s):  
Weak Compactness ◽  
Locally Convex Spaces ◽  
Bounded Sets ◽  
Locally Convex ◽  
Convex Spaces

k-spaces and duals of non-archimedean metrizable locally convex spaces

2018 ◽  
Vol 30 (5) ◽  
pp. 1309-1318
Author(s):  
Maria Cristina Perez-Garcia
Keyword(s):  
New York ◽  
Locally Convex Spaces ◽  
Compact Sets ◽  
Inductive Limits ◽  
Locally Compact ◽  
The One ◽  
Definition Of ◽  
Köthe Sequence Spaces ◽  
Locally Convex ◽  
Convex Spaces

AbstractThe main purpose of this paper is to investigate the non-archimedean counterpart of the classical result stating that the dual of a real or complex metrizable locally convex space, equipped with the locally convex topology of uniform convergence on compact sets, belongs to the topological category formed by the k-spaces. We prove that this counterpart holds when the non-archimedean valued base field {\mathbb{K}} is locally compact, but fails for any non-locally compact {\mathbb{K}}. Here we deal with a topological subcategory, the one formed by the {k_{0}}-spaces, the adequate non-archimedean substitutes for k-spaces. As a product, we complete some of the achievements on the non-archimedean Banach–Dieudonné Theorem presented in [C. Perez-Garcia and W. H. Schikhof, The p-adic Banach–Dieudonné theorem and semi-compact inductive limits, p-adic Functional Analysis (Poznań 1998), Lecture Notes Pure Appl. Math. 207, Dekker, New York 1999, 295–307]. Also, we use our results to construct in a simple way natural examples of k-spaces (which are also {k_{0}}-spaces) whose products are not {k_{0}}-spaces. This in turn improves the, rather involved, example given in [C. Perez-Garcia and W. H. Schikhof, Locally Convex Spaces over non-Archimedean Valued Fields, Cambridge Stud. Adv. Math. 119, Cambridge University Press, Cambridge, 2010] of two {k_{0}}-spaces whose product is not a {k_{0}}-space. Our theory covers an important class of non-archimedean Fréchet spaces, the Köthe sequence spaces, which have a relevant influence on applications such as the definition of a non-archimedean Laplace and Fourier transform.

scholarly journals Использование гомологических методов на базе итерированных спектров в функциональном анализе

2012 ◽  
Author(s):  
E.I. Smirnov
Keyword(s):  
Locally Convex Spaces ◽  
Closed Graph ◽  
Closed Graph Theorem ◽  
Inductive Limits ◽  
Quotient Spaces ◽  
New Concepts ◽  
Inductive And Projective Limits ◽  
Projective Limits ◽  
Locally Convex ◽  
Convex Spaces

We introduce new concepts of functional analysis: Hausdorff spectrum and Hausdorff limit or H-limit of Hausdorff spectrum of locally convex spaces. Particular cases of regular H-limit are projective and inductive limits of separated locally convex spaces. The class of H-spaces contains Frechet spaces and is stable under forming countable inductive and projective limits, closed subspaces and quotient spaces. Moreover, for H-space an unproved variant of the closed graph theorem holds true. Homological methods are used for proving of theorems of vanishing at zero for first derivative of Hausdorff limit functor: Haus1(X)=0.

scholarly journals On sequentially retractive inductive limits

2003 ◽  
Vol 2003 (17) ◽  
pp. 1067-1072
Author(s):  
Armando García
Keyword(s):  
Inductive Limit ◽  
Locally Convex Spaces ◽  
Inductive Limits ◽  
Locally Convex ◽  
Convex Spaces

Every locally complete inductive limit of sequentially complete locally convex spaces, which satisfies Retakh's condition(M)is regular, sequentially complete and sequentially retractive. A quasiconverse for this theorem and a criterion for sequential retractivity of inductive limits of webbed spaces are given.

On the covering of bounded sets in locally convex spaces

1973 ◽  
Vol 6 (4) ◽  
pp. 335-336 ◽  
Author(s):  
A. E. Merzon
Keyword(s):  
Locally Convex Spaces ◽  
Bounded Sets ◽  
Locally Convex ◽  
Convex Spaces

scholarly journals Quasi-bounded sets

1990 ◽  
Vol 13 (3) ◽  
pp. 607-610
Author(s):  
Jan Kucera
Keyword(s):  
Fréchet Space ◽  
Locally Convex Spaces ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Bounded Set ◽  
Bounded Sets ◽  
Locally Convex ◽  
Convex Spaces

It is proved in [1] & [2] that a set bounded in an inductivelimit E=indlim Enof Fréchet spaces is also bounded in someEniffEis fast complete. In the case of arbitrary locally convex spacesEnevery bounded set in a fast completeindlim Enis quasi-bounded in someEn, though it may not be bounded or even contained in anyEn. Every bounded set is quasi-bounded. In a Fréchet space every quasi-bounded set is also bounded.

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