scholarly journals A Study of Approximation Properties in Felbin-Fuzzy Normed Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 161
Author(s):  
Ju Myung Kim ◽  
Keun Young Lee
Keyword(s):  
Approximation Properties ◽  
Normed Spaces ◽  
Dual Problems ◽  
Dual Spaces ◽  
Infinite Sequences

In this paper, approximation properties in Felbin-fuzzy normed spaces are studied. These approximation properties have been recently introduced in Felbin-fuzzy normed spaces. We make topological tools to analyze such approximation properties. We especially develop the representation of dual spaces related to our contexts. By using this representation, we establish characterizations of approximation properties in terms of infinite sequences. Finally, we provide dual problems for approximation properties and their results in our contexts.

scholarly journals Notes on Lipschitz Properties of Nonlinear Scalarization Functions with Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Fang Lu ◽  
Chun-Rong Chen
Keyword(s):  
Lipschitz Constant ◽  
Sufficient Conditions ◽  
Normed Spaces ◽  
Lipschitz Property ◽  
Nonlinear Scalarization ◽  
Dual Spaces ◽  
Lipschitz Constants ◽  
Primal And Dual ◽  
Vector Equilibrium ◽  
The One

Various kinds of nonlinear scalarization functions play important roles in vector optimization. Among them, the one commonly known as the Gerstewitz function is good at scalarizing. In linear normed spaces, the globally Lipschitz property of such function is deduced via primal and dual spaces approaches, respectively. The equivalence of both expressions for globally Lipschitz constants obtained by primal and dual spaces approaches is established. In particular, when the ordering cone is polyhedral, the expression for calculating Lipschitz constant is given. As direct applications of the Lipschitz property, several sufficient conditions for Hölder continuity of both single-valued and set-valued solution mappings to parametric vector equilibrium problems are obtained using the nonlinear scalarization approach.

Approximation properties in fuzzy normed spaces

2016 ◽  
Vol 282 ◽  
pp. 115-130 ◽  
Author(s):  
Keun Young Lee
Keyword(s):  
Approximation Properties ◽  
Normed Spaces

scholarly journals Slice convergence of parametrised sums of convex functions in non-reflexive spaces

1999 ◽  
Vol 60 (3) ◽  
pp. 429-458 ◽  
Author(s):  
Robert Wenczel ◽  
Andrew Eberhard
Keyword(s):  
Convex Functions ◽  
Mosco Convergence ◽  
Normed Spaces ◽  
Unified Framework ◽  
Dual Spaces ◽  
Slice Convergence ◽  
Sequential Continuity ◽  
Kuratowski Convergence ◽  
Primal And Dual ◽  
Fenchel Conjugation

The objectives of this study of slice convergence are two-fold. The first is to derive results regarding the passage of certain semi–convergences through Young–Fenchel conjugation. These semi–convergences arise from the splitting of the usual slice topology in the primal and dual spaces into (non-Hausdorff) topologies: the upper slice topology ; a topology generating a convergence closely resembling the bounded–weak* upper Kuratowski convergence; along with the respective primal and dual lower Kuratowski topologies. This gives rise to topological convergences not reliant on sequentially–based definitions found in many such studies, and associated topological continuity results for conjugation (in normed spaces), in contrast to the usual sequential continuity exhibited by analogues of Mosco convergence. The second objective is to study the passage of slice convergence through addition. Such sum theorems have been derived in other works and we establish previous theorems from a unified framework as well as obtaining a new result.

scholarly journals Characterizations of Rotundity and Smoothness by Approximate Orthogonalities

2016 ◽  
Vol 30 (1) ◽  
pp. 193-201 ◽  
Author(s):  
Tomasz Stypuła ◽  
Paweł Wójcik
Keyword(s):  
Normed Spaces ◽  
Dual Spaces

AbstractIn this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity and smoothness of dual spaces.

scholarly journals Dual problems for weak and quasi approximation properties

2006 ◽  
Vol 321 (2) ◽  
pp. 569-575 ◽  
Author(s):  
Ju Myung Kim
Keyword(s):  
Approximation Properties ◽  
Dual Problems

scholarly journals Approximation Properties in Felbin Fuzzy Normed Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 1003 ◽  
Author(s):  
Ju Myung Kim ◽  
Keun Young Lee
Keyword(s):  
Comparative Study ◽  
Finite Rank ◽  
Approximation Properties ◽  
Bounded Operators ◽  
Normed Spaces ◽  
New Concepts

In this paper, approximation properties in Felbin fuzzy normed spaces are considered. These approximation properties are new concepts in Felbin fuzzy normed spaces. Definitions and examples of such properties are given and we make a comparative study among approximation properties in Bag and Samanta fuzzy normed spaces and Felbin fuzzy normed spaces. We develop the representation of finite rank bounded operators in our context. By using this representation, characterizations of approximation properties are established in Felbin fuzzy normed spaces.

scholarly journals Approximation properties for dual spaces

2003 ◽  
Vol 93 (2) ◽  
pp. 297 ◽  
Author(s):  
Vegard Lima
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Approximation Property ◽  
Extension Operator ◽  
Approximation Properties ◽  
Dual Spaces ◽  
Metric Approximation

We prove that a Banach space $X$ has the metric approximation property if and only if $\mathcal F(Y,X)$ is an ideal in $\mathcal L(Y,X^{**})$ for all Banach spaces $Y$. Furthermore, $X^*$ has the metric approximation property if and only if for all Banach spaces $Y$ and all Hahn-Banach extension operators $\phi : X^* \rightarrow X^{***}$ there exists a Hahn-Banach extension operator $\Phi : {\mathcal F(Y,X)}^* \rightarrow {\mathcal L(Y,X^{**})}^*$ such that $\Phi(x^* \otimes y^{**}) = (\phi x^*) \otimes y^{**}$ for all $x^* \in X^*$ and all $y^{**} \in Y^{**}$. We also prove that $X^*$ has the approximation property if and only if for all Banach spaces $Y$ and all Hahn-Banach extension operators $\phi : X^* \rightarrow X^{***}$ there exists a Hahn-Banach extension operator $\Phi : {\mathcal F(Y,X)}^* \rightarrow {\mathcal W(Y,X^{**})}^*$ such that $\Phi(x^* \otimes y^{**}) = (\phi x^*) \otimes y^{**}$ for all $x^* \in X^*$ and all $y^{**} \in Y^{**}$, which in turn is equivalent to $\mathcal F(Y,\hat{X})$ being an ideal in $\mathcal W(Y,\hat{X}^{**})$ for all Banach spaces $Y$ and all equivalent renormings $\hat{X}$ of $X$.

scholarly journals Dual Spaces and Hahn-Banach Theorem

2014 ◽  
Vol 22 (1) ◽  
pp. 69-77 ◽  
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Extension Theorem ◽  
Normed Spaces ◽  
Linear Spaces ◽  
Dual Spaces ◽  
Basic Properties ◽  
Banach Theorem ◽  
Real Linear

Summary In this article, we deal with dual spaces and the Hahn-Banach Theorem. At the first, we defined dual spaces of real linear spaces and proved related basic properties. Next, we defined dual spaces of real normed spaces. We formed the definitions based on dual spaces of real linear spaces. In addition, we proved properties of the norm about elements of dual spaces. For the proof we referred to descriptions in the article [21]. Finally, applying theorems of the second section, we proved the Hahn-Banach extension theorem in real normed spaces. We have used extensively used [17].

scholarly journals Weak Convergence and Weak Convergence

2015 ◽  
Vol 23 (3) ◽  
pp. 231-241
Author(s):  
Keiko Narita ◽  
Yasunari Shidama ◽  
Noboru Endou
Keyword(s):  
Weak Convergence ◽  
Real Number ◽  
Banach Spaces ◽  
Topological Properties ◽  
Normed Spaces ◽  
Real Numbers ◽  
Dual Spaces ◽  
Sequential Compactness

Abstract In this article, we deal with weak convergence on sequences in real normed spaces, and weak* convergence on sequences in dual spaces of real normed spaces. In the first section, we proved some topological properties of dual spaces of real normed spaces. We used these theorems for proofs of Section 3. In Section 2, we defined weak convergence and weak* convergence, and proved some properties. By RNS_Real Mizar functor, real normed spaces as real number spaces already defined in the article [18], we regarded sequences of real numbers as sequences of RNS_Real. So we proved the last theorem in this section using the theorem (8) from [25]. In Section 3, we defined weak sequential compactness of real normed spaces. We showed some lemmas for the proof and proved the theorem of weak sequential compactness of reflexive real Banach spaces. We referred to [36], [23], [24] and [3] in the formalization.

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