scholarly journals Dual pairs of sequence spaces

2001 ◽  
Vol 28 (1) ◽  
pp. 9-23 ◽  
Author(s):  
Johann Boos ◽  
Toivo Leiger
Keyword(s):  
Sequence Space ◽  
Dual Pair ◽  
General Concept ◽  
Sequence Spaces ◽  
Dual Pairs ◽  
The Third ◽  
Convex Sequence ◽  
Starting Point ◽  
Alpha Beta ◽  
Locally Convex

The paper aims to develop for sequence spacesEa general concept for reconciling certain results, for example inclusion theorems, concerning generalizations of the Köthe-Toeplitz dualsE×(×∈{α,β})combined with dualities(E,G),G⊂E×, and theSAK-property (weak sectional convergence). TakingEβ:={(yk)∈ω:=𝕜ℕ|(ykxk)∈cs}=:Ecs, wherecsdenotes the set of all summable sequences, as a starting point, then we get a general substitute ofEcsby replacingcsby any locally convex sequence spaceSwith sums∈S′(in particular, a sum space) as defined by Ruckle (1970). This idea provides a dual pair(E,ES)of sequence spaces and gives rise for a generalization of the solid topology and for the investigation of the continuity of quasi-matrix maps relative to topologies of the duality(E,Eβ). That research is the basis for general versions of three types of inclusion theorems: two of them are originally due to Bennett and Kalton (1973) and generalized by the authors (see Boos and Leiger (1993 and 1997)), and the third was done by Große-Erdmann (1992). Finally, the generalizations, carried out in this paper, are justified by four applications with results around different kinds of Köthe-Toeplitz duals and related section properties.

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 Domain of the Double Sequential Band Matrix in the Sequence Space

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Havva Nergiz ◽  
Feyzi Başar
Keyword(s):  
Sequence Space ◽  
Sequence Spaces ◽  
Schauder Basis ◽  
Matrix Transformations ◽  
Difference Matrix ◽  
Band Matrix ◽  
Alpha Beta

The sequence space was introduced by Maddox (1967). Quite recently, the domain of the generalized difference matrix in the sequence space has been investigated by Kirişçi and Başar (2010). In the present paper, the sequence space of nonabsolute type has been studied which is the domain of the generalized difference matrix in the sequence space . Furthermore, the alpha-, beta-, and gamma-duals of the space have been determined, and the Schauder basis has been given. The classes of matrix transformations from the space to the spaces ,candc0have been characterized. Additionally, the characterizations of some other matrix transformations from the space to the Euler, Riesz, difference, and so forth sequence spaces have been obtained by means of a given lemma. The last section of the paper has been devoted to conclusion.

scholarly journals Linear functionals on Orlicz sequence spaces without local convexity

1992 ◽  
Vol 15 (2) ◽  
pp. 241-254 ◽  
Author(s):  
Marian Nowak
Keyword(s):  
Sequence Space ◽  
Sequence Spaces ◽  
Local Convexity ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Orlicz Sequence Space ◽  
Orlicz Sequence Spaces ◽  
Locally Convex

The general form of continuous linear functionals on an Orlicz sequence space1ϕ(non-separable and non-locally convex in general) is obtained. It is proved that the spacehϕis anM-ideal in1ϕ.

scholarly journals ON GENERALIZED FIBONACCI DIFFERENCE SPACE DERIVED FROM THE ABSOLUTELY p− SUMMABLE SEQUENCE SPACES

2019 ◽  
pp. 903
Author(s):  
Gülsen Kılınç
Keyword(s):  
Sequence Space ◽  
Bounded Sequence ◽  
Sequence Spaces ◽  
The Other ◽  
Matrix Transformations ◽  
Real Numbers ◽  
Positive Real ◽  
Topological Features ◽  
Summable Sequence ◽  
Alpha Beta

In this study, it is specified \emph{the sequence space} $l\left( F\left( r,s\right),p\right) $, (where $p=\left( p_{k}\right) $ is any bounded sequence of positive real numbers) and researched some algebraic and topological features of this space. Further, $\alpha -,$ $\beta -,$ $\gamma -$ duals and its Schauder Basis are given. The classes of \emph{matrix transformations} from the space $l\left( F\left( r,s\right) ,p\right) $ to the spaces $l_{\infty },c,$ and $% c_{0}$ are qualified. Additionally, acquiring qualifications of some other \emph{matrix transformations} from the space $l\left( F\left( r,s\right) ,p\right) $ to the \emph{Euler, Riesz, difference}, etc., \emph{sequence spaces} is the other result of the paper.

scholarly journals SOME GENERALIZED FIBONACCI DIFFERENCE SPACES DEFINED BY A SEQUENCE OF MODULUS FUNCTIONS

2017 ◽  
pp. 095 ◽  
Author(s):  
Gülsen Kılınç ◽  
Murat Candan
Keyword(s):  
Real Number ◽  
Sequence Space ◽  
Sequence Spaces ◽  
Inclusion Relations ◽  
Alpha Beta

This paper submits the sequence space $l\left( \widehat{F}\left( r,s\right),\mathcal{F},p,u\right) $ and $l_{\infty }\left( \widehat{F}\left(r,s\right) ,\mathcal{F},p,u\right) $of non-absolute type under the domain ofthe matrix$\widehat{\text{ }F}\left( r,s\right) $ constituted by usingFibonacci sequence and non-zero real number $r$, $s$ and a sequence ofmodulus functions. We study some inclusion relations, topological andgeometric properties of these spaceses. Further, we give the $\alpha $- $%\beta $- and $\gamma $-duals of said sequence spaces and characterization ofthe classes $\left( l\left( \widehat{F}\left( r,s\right) ,\mathcal{F}%,p,u\right) ,X\right) $ and $\left( l_{\infty }\left( \widehat{F}\left(r,s\right) ,\mathcal{F},p,u\right) ,X\right) $.

An Abstract Concept of the Sum of a Numerical Series

1970 ◽  
Vol 22 (4) ◽  
pp. 863-874 ◽  
Author(s):  
William H. Ruckle
Keyword(s):  
Topological Space ◽  
Sequence Space ◽  
Dual Space ◽  
Linear Topological Space ◽  
Sequence Spaces ◽  
Abstract Concept ◽  
Numerical Series ◽  
Locally Convex ◽  
Theory Of Duality

Our aim in this paper, generally stated, is to formulate an abstract concept of the notion of the sum of a numerical series. More particularly, it is a study of the type of sequence space called “sum space”. The idea of sum space arose in connection with two distinct problems.1.1 The Köthe-Toeplitz dual of a sequence space T consists of all sequences t such that st ∈ l1 (absolutely summable sequences) for each s∈T. It is known that if cs or bs is used in place of l1, an analogous theory of duality for sequence spaces can be developed (cf. [2]). What other spaces of sequences can play a rôle analogous to l1? This problem is treated in [6].1.2. Let {xn, fn} be a complete biorthogonal sequence in (X, X*), where X is a locally convex linear topological space and X* is its topological dual space.

scholarly journals Kants Konzeption kosmologischer Freiheit – ein metaphysischer Rest?

2017 ◽  
Vol 2 (2) ◽  
pp. 179
Author(s):  
Christian Krijnen
Keyword(s):  
18Th Century ◽  
General Concept ◽  
Transcendental Philosophy ◽  
Practical Philosophy ◽  
Philosophical Thought ◽  
The Third ◽  
Starting Point ◽  
Metaphysical Tradition

The German idealists were of the opinion that Kant’s transcendental turn has unfettered a revolution in philosophical thought that needs to be completed by addressing critically the presuppositions or ‘foundations’ of Kant’s philosophy itself. To these presuppositions belong Kant’s architectonic of reason in general and the role the concept of freedom has within it in particular. It is shown that Kant’s cosmological or transcendental freedom does not so much establishes a secure starting point for further elaborations within the realm of practical philosophy but should primarily be taken as a problem in itself. By doing so, it becomes clear that the profile of Kant’s critical conception of freedom as well as that of the third antinomy heavily draw upon the German metaphysical tradition of the 18th century. By considering Kant’s conception of cosmological freedom in the context of the discussions of his age, several preliminaries and non justified constellations come into view. From Hegel’s perspective, they cannot be justified; rather, getting to the bottom of them transcendentally leads to a more general concept of freedom than Kant’s. The consequences of all this are illustrated by revisiting the transcendental philosophy of Bruno Bauch, probably the best neo-Kantian Kant specialist.

scholarly journals Topological properties and matrix transformations of certain ordered generalized sequence spaces

1995 ◽  
Vol 18 (2) ◽  
pp. 341-356 ◽  
Author(s):  
Manjul Gupta ◽  
Kalika Kaushal
Keyword(s):  
Sequence Space ◽  
Topological Structure ◽  
Necessary Conditions ◽  
Riesz Space ◽  
Linear Operators ◽  
Sequence Spaces ◽  
Matrix Transformations ◽  
Topological Properties ◽  
Vector Valued ◽  
Locally Convex

In this note, we carry out investigations related to the mixed impact of ordering and topological structure of a locally convex solid Riesz space(X,τ)and a scalar valued sequence spaceλ, on the vector valued sequence spaceλ(X)which is formed and topologized with the help ofλandX, and vice versa. Besides,we also characterizeo-matrix transformations fromc(X),ℓ∞(X)to themselves,cs(X)toc(X)and derive necessary conditions for a matrix of linear operators to transformℓ1(X)into a simple ordered vector valued sequence spaceΛ(X).

Teologiczne znaczenie osoby i natury w świetle chrystologii Soboru Chalcedońskiego

2018 ◽  
Vol 31 (4) ◽  
pp. 146-165
Author(s):  
Giacomo Calore
Keyword(s):  
The Third ◽  
Starting Point

Council of Chalcedon is an actual closing point for Christology and a starting point for anthropology. Behind the teachings of the Council of Chalcedon,together with later clarifications added by the Second and the Third Councils of Constantinople, there were centuries of dispute between the School of Alexandria and the School of Antioch about the person and natures of Christ (4th/5th – 7th centuries). Therefore the light shed on the man by patristic Christology concerns understanding of his being a person and his nature. The analysis of the Council’s teachings of faith shows that these two concepts belong to two different areas which means that every man, following the man Jesus, is a person whose dignity is on a different level than his natural features (mind, will, consciousness, etc.) – in other words, it originates from transcendence. Simultaneously, person is a relational reality because it puts a man in a relation with God in which the nature can be improved, the nature whose essence – since it was adopted by Logos – is to be capax Dei, or ability to grow in following Christ.

Law and Economics

2020 ◽  
pp. 84-102
Author(s):  
Daniel B. Kelly
Keyword(s):  
Economic Analysis ◽  
First Generation ◽  
Law And Economics ◽  
Private Law ◽  
Foreseeable Future ◽  
Business Associations ◽  
The Third ◽  
Starting Point ◽  
The Law ◽  
The Impact

This chapter analyzes how law and economics influences private law and how (new) private law is influencing law and economics. It focuses on three generation or “waves” within law and economics and how they approach private law. In the first generation, many scholars took the law as a starting point and attempted to use economic insights to explain, justify, or reform legal doctrines, institutions, and structures. In the second generation, the “law” at times became secondary, with more focus on theory and less focus on doctrines, institutions, and structures. But this generation also relied increasingly on empirical analysis. In the third generation, which includes scholars in the New Private Law (NPL), there has been a resurgence of interest in the law and legal institutions. To be sure, NPL scholars analyze the law using various approaches, with some more and some less predisposed to economic analysis. However, economic analysis will continue to be a major force on private law, including the New Private Law, for the foreseeable future. The chapter considers three foundational private law areas: property, contracts, and torts. For each area, it discusses the major ideas that economic analysis has contributed to private law, and surveys contributions of the NPL. The chapter also looks at the impact of law and economics on advanced private law areas, such as business associations, trusts and estates, and intellectual property.

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