scholarly journals A Remark on the Closed Graph Theorem in Locally Convex Vector Spaces.

1966 ◽  
Vol 19 ◽  
pp. 54 ◽  
Author(s):  
Arne Persson
Keyword(s):  
Vector Spaces ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Locally Convex

scholarly journals On The Closed Graph Theorem

1956 ◽  
Vol 3 (1) ◽  
pp. 9-12 ◽  
Author(s):  
Alex. P. Robertson ◽  
Wendy Robertson
Keyword(s):  
Functional Analysis ◽  
Banach Spaces ◽  
Convex Space ◽  
Vector Spaces ◽  
Complex Field ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Topological Vector Spaces ◽  
Locally Convex

The closed graph theorem is one of the deeper results in the theory of Banach spaces and one of the richest in its applications to functional analysis. This note contains an extension of the theorem to certain classes of topological vector spaces. For the most part, we use the terminology and notation of N. Bourbaki [1], contracting “locally convex topological vector space over the real or complex field” to “convex space”; here we confine ourselves to convex spaces.

scholarly journals Non-archimedean closed graph theorems

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 3933-3945
Author(s):  
Toivo Leiger
Keyword(s):  
Final Section ◽  
Bounded Sequence ◽  
Bounded Subset ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Linear Maps ◽  
Local Convex ◽  
Locally Convex ◽  
Convex Spaces

We consider linear maps T: X ? Y, where X and Y are polar local convex spaces over a complete non-archimedean field K. Recall that X is called polarly barrelled, if each weakly* bounded subset in the dual X0 is equicontinuous. If in this definition weakly* bounded subset is replaced by weakly* bounded sequence or sequence weakly* converging to zero, then X is said to be l?-barrelled or c0-barrelled, respectively. For each of these classes of locally convex spaces (as well as the class of spaces with weakly* sequentially complete dual) as domain class, the maximum class of range spaces for a closed graph theorem to hold is characterized. As consequences of these results, we obtain non-archimedean versions of some classical closed graph theorems. The final section deals with the necessity of the above-named barrelledness-like properties in closed graph theorems. Among others, counterparts of the classical theorems of Mahowald and Kalton are proved.

scholarly journals Baire-Type Properties in Metrizable c0(Ω, X)

Axioms ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 6
Author(s):  
Salvador López-Alfonso ◽  
Manuel López-Pellicer ◽  
Santiago Moll-López
Keyword(s):  
Normed Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Metrizable Space ◽  
Complex Numbers ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Open Problems ◽  
Locally Convex

Ferrando and Lüdkovsky proved that for a non-empty set Ω and a normed space X, the normed space c0(Ω,X) is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered Baire-like. When X is a metrizable locally convex space, with an increasing sequence of semi-norms .n∈N defining its topology, then c0(Ω,X) is the metrizable locally convex space over the field K (of the real or complex numbers) of all functions f:Ω→X such that for each ε>0 and n∈N the set ω∈Ω:f(ω)n>ε is finite or empty, with the topology defined by the semi-norms fn=supf(ω)n:ω∈Ω, n∈N. Kąkol, López-Pellicer and Moll-López also proved that the metrizable space c0(Ω,X) is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p if and only if X is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p. The main result of this paper is that the metrizable c0(Ω,X) is baireled if and only if X is baireled, and its proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented.

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.

Some forms of the closed graph theorem

1971 ◽  
Vol 70 (3) ◽  
pp. 401-408 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Linear Mapping ◽  
Locally Convex Spaces ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Locally Convex ◽  
Convex Spaces

In this paper we shall establish some forms of the closed graph theorem for locally convex spaces, using the approach of Pták(17). Our interest is in classifying pairs of locally convex spaces (E, F) which have the property that every closed graph linear mapping T: E → F is continuous; if (E, F) has this property then we shall say that (E, F) is in the class ℒ if is a particular class of locally convex spaces then ℒ() is the class of all E such that (E, F)∈ℒ for all F ∈ .

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 Ultracomplete convergence vector spaces and the closed graph theorem

2001 ◽  
Vol 111 (1-2) ◽  
pp. 59-69 ◽  
Author(s):  
R. Beattie ◽  
H.-P. Butzmann
Keyword(s):  
Vector Spaces ◽  
Closed Graph ◽  
Graph Theorem ◽  
Closed Graph Theorem

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.

scholarly journals -spaces and the closed-graph theorem

1968 ◽  
Vol 16 (2) ◽  
pp. 89-99 ◽  
Author(s):  
S. O. Iyahen
Keyword(s):  
Topological Spaces ◽  
Topological Groups ◽  
Closed Graph ◽  
Range Space ◽  
Graph Theorem ◽  
Closed Graph Theorem ◽  
Complete Space ◽  
Locally Convex

The problem considered in this paper is that of finding conditions on a range space such that the closed-graph theorem holds for linear mappings from a class of linear topological spaces. The concept of a -space, which is a result of this investigation, is meaningful for commutative topological groups but we limit our consideration in this paper to linear topological spaces. On restricting ourselves to locally convex linear topological spaces, we see that the notion of a -space is an extension of the powerful idea of a B-complete space.

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