Barrelled spaces and the closed graph theorem

2006 ◽  
pp. 41-46
Keyword(s):  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Barrelled Spaces

scholarly journals On the closed graph theorem

1976 ◽  
Vol 17 (2) ◽  
pp. 89-97 ◽  
Author(s):  
J. O. Popoola ◽  
I. Tweddle
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Cardinal Number ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Infinite Dimensional ◽  
Barrelled Spaces ◽  
Natural Sense ◽  
Locally Convex

Our main purpose is to describe those separated locally convex spaces which can serve as domain spaces for a closed graph theorem in which the range space is an arbitrary Banach space of (linear) dimension at most c, the cardinal number of the real line R. These are the δ-barrelled spaces which are considered in §4. Many of the standard elementary Banach spaces, including in particular all separable ones, have dimension at most c. Also it is known that an infinite dimensional Banach space has dimension at least c (see e.g. [8]). Thus if we classify Banach spaces by dimension we are dealing, in a natural sense, with the first class which contains infinite dimensional spaces.

scholarly journals Countably quasi-suprabarrelled spaces

1993 ◽  
Vol 48 (1) ◽  
pp. 1-6
Author(s):  
J.C. Ferrando ◽  
L.M. Sánchez Ruiz
Keyword(s):  
Locally Convex Spaces ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Barrelled Spaces ◽  
Permanence Properties ◽  
Locally Convex ◽  
Convex Spaces

In this paper we obtain some permanence properties of a class of locally convex spaces located between quasi-suprabarrelled spaces and quasi-totally barrelled spaces, for which a closed graph theorem is given.

scholarly journals Quasi-suprabarrelled spaces

1989 ◽  
Vol 46 (1) ◽  
pp. 137-145 ◽  
Author(s):  
J. C. Ferrando ◽  
M. López-Pellicer
Keyword(s):  
Dense Subspace ◽  
Proper Class ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Convex Topology ◽  
Barrelled Spaces ◽  
Permanence Properties ◽  
Locally Convex Topology ◽  
Locally Convex

AbstractIn this paper a proper class of barrelled spaces which strictly contains the suprabarrelled spaces is considered. A closed graph theorem and some permanence properties are given. This allows us to prove the necessity of a condition of a theorem of S. A. Saxon and P. P. Narayanaswami by constructing an example of a non-suprabarrelled Baire-like space which is a dense subspace of a Fréchet space and is not an (LF)-space under any strong locally convex topology.

B(?)-spaces and the closed graph theorem

1964 ◽  
Vol 153 (4) ◽  
pp. 293-298 ◽  
Author(s):  
Taqdir Husain
Keyword(s):  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem

scholarly journals The closed graph theorem without category

1987 ◽  
Vol 36 (2) ◽  
pp. 283-287 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Baire Category ◽  
Baire Category Theorem ◽  
Normed Spaces ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Category Theorem

We show that a diagonal theorem of P. Antosik can be used to give a proof of the Closed Graph Theorem for normed spaces which does not depend upon the Baire Category Theorem.

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