scholarly journals Some Properties oflp(A,X)Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaohong Fu ◽  
Songxiao Li
Keyword(s):  
Simple Proof ◽  
Normed Space ◽  
Convex Space ◽  
Unconditional Basis ◽  
Locally Convex Space ◽  
Locally Convex

We provide a representation of elements of the spacelp(A,X)for a locally convex spaceXand1≤p<∞and determine its continuous dual for normed spaceXand1<p<∞. In particular, we study the extension and characterization of isometries onlp(N,X)space, whenXis a normed space with an unconditional basis and with a symmetric norm. In addition, we give a simple proof of the main result of G. Ding (2002).

scholarly journals In what spaces is every closed normal cone regular?

1970 ◽  
Vol 17 (2) ◽  
pp. 121-125 ◽  
Author(s):  
C. W. McArthur
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Unconditional Basis ◽  
Normal Cone ◽  
Closed Subspace ◽  
Locally Convex Space ◽  
Regular Part ◽  
Fréchet Space ◽  
Sequential Completeness ◽  
Locally Convex

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

Some criteria for nuclearity

1986 ◽  
Vol 100 (1) ◽  
pp. 151-159 ◽  
Author(s):  
M. A. Sofi
Keyword(s):  
Sequence Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Sequence Spaces ◽  
Bilinear Forms ◽  
Simple Formulation ◽  
Normal Topology ◽  
Nuclear Spaces ◽  
Locally Convex

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

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 Coincidence points and invariant approximation results on q-normed spaces

Filomat ◽  
2007 ◽  
Vol 21 (2) ◽  
pp. 45-53 ◽  
Author(s):  
Kumar Nashine
Keyword(s):  
Normed Space ◽  
Convex Space ◽  
Nonexpansive Mappings ◽  
Locally Convex Space ◽  
Normed Spaces ◽  
Coincidence Points ◽  
Approximation Result ◽  
Invariant Approximation ◽  
Locally Convex

We present coincidence points results for multivalued /- nonexpansive mappings in the setting of nonstarshaped domain of q-normed space which is not necessarily a locally convex space. As application, an invariant approximation result is also obtained. Our results improve and extend the results of Bano, Khan and Latif [1], Hussain [4], Latif and Tweddle [7], Rhoades [11], Sahab, Khan and Sessa [13], Shahzad [14] and Singh [15] in the setting of nonstarshaped domain. .

scholarly journals Quotient bounded elements in locally convex algebras

1982 ◽  
Vol 33 (1) ◽  
pp. 102-113 ◽  
Author(s):  
Subhash J. Bhatt
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Spatial Characterization ◽  
Measurable Functions ◽  
Locally Convex ◽  
Locally Convex Algebra ◽  
Continuous Linear Operators

AbstractThe quotient bounded and the universally bounded elements in a calibrated locally convex algebra are defined and studied. In the case of a generalized B*-algebra A, they are shown to form respectively b* and B*-algebras, both dense in A. An internal spatial characterization of generalized B*-algebras is obtained. The concepts are illustrated with the help of examples of algebras of measurable functions and of continuous linear operators on a locally convex space.

scholarly journals Countable enlargements of norm topologies and the quasidistinguished property

1993 ◽  
Vol 35 (2) ◽  
pp. 235-238 ◽  
Author(s):  
F. X. Catalan ◽  
I. Tweddle
Keyword(s):  
General Result ◽  
Normed Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Bounded Subset ◽  
Normed Spaces ◽  
Mackey Topology ◽  
Open Question ◽  
Locally Convex

Let E be a Hausdorff locally convex space with continuous dual E1 and let M be a subspace of the algebraic dual E* such that M ∩ E1 = {0} and dim M = ℵ0. In the terminology of [4] the Mackey topology τ(E, E1 + M) is called a countable enlargement of τ(E, E1). There has been some interest in the question of when barrelledness is preserved under countable enlargements (see [4], [5], [6], [8], [9]). In this note we are concerned with the preservation of the quasidistinguished property for normed spaces under countable enlargements; this was posed as on open question by B. Tsirulnikov in [7]. According to [7] a Hausdorff locally convex space E is quasidistinguished if every bounded subset of its completion Ê is contained in the completion of a bounded subset of Ê (equivalently, in the closure in Ê of a bounded subset of E). Any normed space is clearly quasidistinguished and remains so under a finite enlargement (dim M < χ0) since the enlarged topology is normable. (See the Main Theorem of [7] for a general result on the preservation of the quasidistinguished property under finite enlargements.) We shall write QDCE for a countable enlargement which preserves the quasidistinguished property.

scholarly journals Quasidistinguished countable enlargements of normed spaces

1996 ◽  
Vol 38 (1) ◽  
pp. 65-68
Author(s):  
S. A. Saxon ◽  
L. M. Sànchez Ruiz
Keyword(s):  
Normed Space ◽  
Convex Space ◽  
Dimensional Subspace ◽  
Negative Answer ◽  
Locally Convex Space ◽  
Bounded Subset ◽  
Normed Spaces ◽  
Mackey Topology ◽  
Metrizable Spaces ◽  
Locally Convex

If E is a Hausdorff locally convex space and M is an -dimensional subspace of the algebraic dual E* that is transverse to the continuous dual E′, then, according to [7], the Mackey topology τ(E, E′ + M) is a countable enlargement (CE) of τ(E, E′) [or of E]. Much is still unknown as to when CEs preserve barrelledness (cf. [14]). E is quasidistinguished (QD) if each bounded subset of the completion Ê is contained in the completion of a bounded subset of E [12]. Clearly, each normed space is QD, and Tsirulnikov [12] asked if each CE of a normed space must be a QDCE, i.e., must preserve the QD property. Since CEs preserve metrizability (but not normability), her question was whether metrizable spaces so obtained must be QD, and was moderated by Amemiya's negative answer (cf. [5, p. 404]) to Grothendieck's query, who had asked if all metrizable spaces are QD, having proved the separable ones are [4].

scholarly journals A maximum principle

1979 ◽  
Vol 28 (1) ◽  
pp. 23-26
Author(s):  
Kung-Fu Ng
Keyword(s):  
Maximum Principle ◽  
Extreme Point ◽  
Convex Space ◽  
Maximal Element ◽  
Locally Convex Space ◽  
Compact Set ◽  
Natural Manner ◽  
Upper Semicontinuous ◽  
Nonempty Compact ◽  
Locally Convex

AbstractLet K be a nonempty compact set in a Hausdorff locally convex space, and F a nonempty family of upper semicontinuous convex-like functions from K into [–∞, ∞). K is partially ordered by F in a natural manner. It is shown among other things that each isotone, upper semicontinuous and convex-like function g: K → [ – ∞, ∞) attains its K-maximum at some extreme point of K which is also a maximal element of K.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 46 A 40.

scholarly journals Strictly barrelled disks in inductive limits of quasi-(LB)-spaces

1996 ◽  
Vol 19 (4) ◽  
pp. 727-732
Author(s):  
Carlos Bosch ◽  
Thomas E. Gilsdorf
Keyword(s):  
Convex Space ◽  
Inductive Limit ◽  
Linear Span ◽  
Locally Convex Space ◽  
Minkowski Functional ◽  
Closed Graph ◽  
Inductive Limits ◽  
Locally Convex ◽  
Barrelled Space

A strictly barrelled diskBin a Hausdorff locally convex spaceEis a disk such that the linear span ofBwith the topology of the Minkowski functional ofBis a strictly barrelled space. Valdivia's closed graph theorems are used to show that closed strictly barrelled disk in a quasi-(LB)-space is bounded. It is shown that a locally strictly barrelled quasi-(LB)-space is locally complete. Also, we show that a regular inductive limit of quasi-(LB)-spaces is locally complete if and only if each closed bounded disk is a strictly barrelled disk in one of the constituents.

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