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Journal articles on the topic 'Normed linear spaces and Banach spaces'

Author: Grafiati

Published: 4 June 2021

Last updated: 31 July 2024

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1

Barnes, Benedict, I. A. Adjei, S. K. Amponsah, and E. Harris. "Product-Normed Linear Spaces." European Journal of Pure and Applied Mathematics 11, no. 3 (July 31, 2018): 740–50. http://dx.doi.org/10.29020/nybg.ejpam.v11i3.3284.

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In this paper, both the product-normed linear space $P-NLS$ (product-Banach space) and product-semi-normed linear space (product-semi-Banch space) are introduced. These normed linear spaces are endowed with the first and second product inequalities, which have a lot of applications in linear algebra and differential equations. In addition, we showed that $P-NLS$ admits functional properties such as completeness, continuity and the fixed point.

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2

Ettayb, J. "Some results on ultrametric 2-normed spaces." Researches in Mathematics 32, no. 1 (July 8, 2024): 45. http://dx.doi.org/10.15421/242404.

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In this paper, we study the ultrametric 2-normed spaces and the ultrametric 2-Banach spaces. In particular, we establish some results on Cauchy sequences in ultrametric 2-normed spaces. Also, we introduce and study the notion of bounded linear 2-functionals on ultrametric 2-Banach spaces and we give some of its properties. On the other hand, the new norm on the ultrametric 2-normed space is constructed. The concepts of closed operators between ultrametric 2-normed spaces and $b$-linear functionals in ultrametric 2-normed spaces are introduced. Finally, a necessary and sufficient condition for a linear operator to be closed in terms of its graph is proved and some results on bounded $b$-linear functionals in ultrametric 2-normed spaces are given.

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3

Narita, Keiko, Noboru Endou, and Yasunari Shidama. "Bidual Spaces and Reflexivity of Real Normed Spaces." Formalized Mathematics 22, no. 4 (December 1, 2014): 303–11. http://dx.doi.org/10.2478/forma-2014-0030.

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Summary In this article, we considered bidual spaces and reflexivity of real normed spaces. At first we proved some corollaries applying Hahn-Banach theorem and showed related theorems. In the second section, we proved the norm of dual spaces and defined the natural mapping, from real normed spaces to bidual spaces. We also proved some properties of this mapping. Next, we defined real normed space of R, real number spaces as real normed spaces and proved related theorems. We can regard linear functionals as linear operators by this definition. Accordingly we proved Uniform Boundedness Theorem for linear functionals using the theorem (5) from [21]. Finally, we defined reflexivity of real normed spaces and proved some theorems about isomorphism of linear operators. Using them, we proved some properties about reflexivity. These formalizations are based on [19], [20], [8] and [1].

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4

Bouadjila, K., A. Tallab, and E. Dahia. "Banach-Steinhaus theorem for linear relations on asymmetric normed spaces." Carpathian Mathematical Publications 14, no. 1 (June 30, 2022): 230–37. http://dx.doi.org/10.15330/cmp.14.1.230-237.

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We study the continuity of linear relations defined on asymmetric normed spaces with values in normed spaces. We give some geometric charactirization of these mappings. As an application, we prove the Banach-Steinhaus theorem in the framework of asymmetric normed spaces.

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5

Narita, Keiko, Noboru Endou, and Yasunari Shidama. "Dual Spaces and Hahn-Banach Theorem." Formalized Mathematics 22, no. 1 (March 30, 2014): 69–77. http://dx.doi.org/10.2478/forma-2014-0007.

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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].

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6

Nakasho, Kazuhisa. "Bilinear Operators on Normed Linear Spaces." Formalized Mathematics 27, no. 1 (April 1, 2019): 15–23. http://dx.doi.org/10.2478/forma-2019-0002.

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Summary The main aim of this article is proving properties of bilinear operators on normed linear spaces formalized by means of Mizar [1]. In the first two chapters, algebraic structures [3] of bilinear operators on linear spaces are discussed. Especially, the space of bounded bilinear operators on normed linear spaces is developed here. In the third chapter, it is remarked that the algebraic structure of bounded bilinear operators to a certain Banach space also constitutes a Banach space. In the last chapter, the correspondence between the space of bilinear operators and the space of composition of linear opearators is shown. We referred to [4], [11], [2], [7] and [8] in this formalization.

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7

Alberto De Bernardi, Carlo, and Libor Veselý. "Tilings of Normed Spaces." Canadian Journal of Mathematics 69, no. 02 (April 2017): 321–37. http://dx.doi.org/10.4153/cjm-2015-057-3.

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Abstract By a tiling of a topological linear space X, we mean a covering of X by at least two closed convex sets, called tiles, whose nonempty interiors are pairwise disjoint. Study of tilings of infinite dimensional spaceswas initiated in the 1980's with pioneer papers by V. Klee. We prove some general properties of tilings of locally convex spaces, and then apply these results to study the existence of tilings of normed and Banach spaces by tiles possessing certain smoothness or rotundity properties. For a Banach space X, our main results are the following. (i) X admits no tiling by Fréchet smooth bounded tiles. (ii) If X is locally uniformly rotund (LUR), it does not admit any tiling by balls. (iii) On the other hand, some spaces, г uncountable, do admit a tiling by pairwise disjoint LUR bounded tiles.

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8

Gupta, Sahil, and T. D. Narang. "On strong proximinality in normed linear spaces." Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica 70, no. 1 (July 4, 2016): 19. http://dx.doi.org/10.17951/a.2016.70.1.19.

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The paper deals with strong proximinality in normed linear spaces. It is proved that in a compactly locally uniformly rotund Banach space, proximinality, strong proximinality, weak approximative compactness and approximative compactness are all equivalent for closed convex sets. How strong proximinality can be transmitted to and from quotient spaces has also been discussed.

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9

El-Shobaky, Entisarat, Sahar Mohammed Ali, and Wataru Takahashi. "On projection constant problems and the existence of metric projections in normed spaces." Abstract and Applied Analysis 6, no. 7 (2001): 401–11. http://dx.doi.org/10.1155/s1085337501000732.

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We give the sufficient conditions for the existence of a metric projection onto convex closed subsets of normed linear spaces which are reduced conditions than that in the case of reflexive Banach spaces and we find a general formula for the projections onto the maximal proper subspaces of the classical Banach spacesl p,1≤p<∞andc 0. We also give the sufficient and necessary conditions for an infinite matrix to represent a projection operator froml p,1≤p<∞orc 0onto anyone of their maximal proper subspaces.

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10

Nurnugroho, Burhanudin Arif, Supama Supama, and A. Zulijanto. "Operator Linear-2 Terbatas pada Ruang Bernorma-2 Non-Archimedean." Jurnal Fourier 8, no. 2 (October 31, 2019): 43–50. http://dx.doi.org/10.14421/fourier.2019.82.43-50.

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Di dalam paper ini dikonstruksikan operator linear-2 terbatas dari X2 ke Y , dengan X ruang bernorma-2 non-Archimedean dan ruang bernorma non-Archimedean. Di dalam paper ini ditunjukan bahwa himpunan semua operator linear-2 terbatas dari X2 to Y , ditulis B(X2, Y) merupakan ruang bernorma non-Archimedean. Selanjutnya, ditunjukan bahwa B(X2, Y), apabila Y ruang Banach non-Archimedean. [In this paper we construct bounded 2-linear operators from X2 to Y, where X is non-Archimedean 2-normed spaces and is a non-Archimedean-normed space. We prove that the set of all bounded 2-linear operators from X2 to Y , denoted by B(X2, Y) is a non-Archimedean normed spaces. Furthermore, we show that B(X2, Y) is a non-Archimedean Banach normed space, whenever Y is a non-Archimedean Banach space.]

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11

Alegre, Carmen, and Salvador Romaguera. "The Hahn-Banach Extension Theorem for Fuzzy Normed Spaces Revisited." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/151472.

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This paper deals with fuzzy normed spaces in the sense of Cheng and Mordeson. We characterize fuzzy norms in terms of ascending and separating families of seminorms and prove an extension theorem for continuous linear functionals on a fuzzy normed space. Our result generalizes the classical Hahn-Banach extension theorem for normed spaces.

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12

Du, Dandan, Asif Ahmad, Anwarud Din, and Yongjin Li. "Some Moduli of Angles in Banach Spaces." Mathematics 10, no. 16 (August 17, 2022): 2965. http://dx.doi.org/10.3390/math10162965.

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In this paper, we mainly discuss the angle modulus of convexity δXa(ϵ) and the angle modulus of smoothness ρXa(ϵ) in a real normed linear space X, which are closely related to the classical modulus of convexity δX(ϵ) and the modulus of smoothness ρX(ϵ). Some geometric properties of the two moduli were investigated. In particular, we obtained a characterization of uniform non-squareness in terms of ρXa(1). Meanwhile, we studied the relationships between δXa(ϵ), ρXa(ϵ) and other geometric constants of real normed linear spaces through some equalities and inequalities. Moreover, these two coefficients were computed for some concrete spaces.

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13

Abbas, Mujahid, Basit Ali, and Salvador Romaguera. "Generalized Contraction and Invariant Approximation Results on Nonconvex Subsets of Normed Spaces." Abstract and Applied Analysis 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/391952.

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Wardowski (2012) introduced a new type of contractive mapping and proved a fixed point result in complete metric spaces as a generalization of Banach contraction principle. In this paper, we introduce a notion of generalizedF-contraction mappings which is used to prove a fixed point result for generalized nonexpansive mappings on star-shaped subsets of normed linear spaces. Some theorems on invariant approximations in normed linear spaces are also deduced. Our results extend, unify, and generalize comparable results in the literature.

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14

Blanco, A., and A. Turnšek. "On maps that preserve orthogonality in normed spaces." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 4 (August 2006): 709–16. http://dx.doi.org/10.1017/s0308210500004674.

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We show that every orthogonality-preserving linear map between normed spaces is a scalar multiple of an isometry. Using this result, we generalize Uhlhorn's version of Wigner's theorem on symmetry transformations to a wide class of Banach spaces.

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15

BARNES, BRUCE A. "BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY." Glasgow Mathematical Journal 49, no. 1 (January 2007): 145–54. http://dx.doi.org/10.1017/s0017089507003503.

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Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

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16

Nakasho, Kazuhisa, and Yasunari Shidama. "Continuity of Multilinear Operator on Normed Linear Spaces." Formalized Mathematics 27, no. 1 (April 1, 2019): 61–65. http://dx.doi.org/10.2478/forma-2019-0006.

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Summary In this article, various definitions of contuity of multilinear operators on normed linear spaces are discussed in the Mizar formalism [4], [1] and [2]. In the first chapter, several basic theorems are prepared to handle the norm of the multilinear operator, and then it is formalized that the linear space of bounded multilinear operators is a complete Banach space. In the last chapter, the continuity of the multilinear operator on finite normed spaces is addressed. Especially, it is formalized that the continuity at the origin can be extended to the continuity at every point in its whole domain. We referred to [5], [11], [8], [9] in this formalization.

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17

Dutta, Hemen. "ON n-NORMED LINEAR SPACE VALUED STRONGLY (C,1)-SUMMABLE DIFFERENCE SEQUENCES." Asian-European Journal of Mathematics 03, no. 04 (December 2010): 565–75. http://dx.doi.org/10.1142/s1793557110000441.

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In this article we extend the notion of famous strongly ( C ,1)-summable or strongly Cesàro summable real sequences to n-normed linear space valued difference sequences. Consequently we introduce the notions of n-normed linear space (n-nls) valued strongly Cesàro [Formula: see text]-summable, strongly Cesàro [Formula: see text]-null and strongly Cesàro [Formula: see text]-bounded sequences. Further we extend and investigate the notion of n-norm and derived (n - l) - norms, for all l = 1, 2, …, n - 1 on the spaces of these three types of sequences. We also prove the Fixed point theorem for these spaces, which are n-Banach spaces under certain conditions and compute the n-isometrically isomorphic spaces. This article also introduces an idea for constructing n-norm on spaces of n-nls valued summable difference sequences.

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18

Ostrovskii, M. I. "Auerbach Bases and Minimal Volume Sufficient Enlargements." Canadian Mathematical Bulletin 54, no. 4 (December 1, 2011): 726–38. http://dx.doi.org/10.4153/cmb-2011-043-3.

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AbstractLet BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection P: Y → X such that P(BY ) ⊂ A. Each finite dimensional normed space has a minimal-volume sufficient enlargement that is a parallelepiped; some spaces have “exotic” minimal-volume sufficient enlargements. The main result of the paper is a characterization of spaces having “exotic” minimal-volume sufficient enlargements in terms of Auerbach bases.

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19

Anatriello, Giuseppina, Alberto Fiorenza, and Giovanni Vincenzi. "Banach function norms via Cauchy polynomials and applications." International Journal of Mathematics 26, no. 10 (September 2015): 1550083. http://dx.doi.org/10.1142/s0129167x15500834.

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Let X1,…,Xk be quasinormed spaces with quasinorms | ⋅ |j, j = 1,…,k, respectively. For any f = (f1,⋯,fk) ∈ X1 ×⋯× Xk let ρ(f) be the unique non-negative root of the Cauchy polynomial [Formula: see text]. We prove that ρ(⋅) (which in general cannot be expressed by radicals when k ≥ 5) is a quasinorm on X1 ×⋯× Xk, which we call root quasinorm, and we find a characterization of this quasinorm as limit of ratios of consecutive terms of a linear recurrence relation. If X1,…,Xk are normed, Banach or Banach function spaces, then the same construction gives respectively a normed, Banach or a Banach function space. Norms obtained as roots of polynomials are already known in the framework of the variable Lebesgue spaces, in the case of the exponent simple function with values 1,…,k. We investigate the properties of the root quasinorm and we establish a number of inequalities, which come from a rich literature of the past century.

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20

Rhoades, B. E., S. sessa, M. S. Khan, and M. Swaleh. "On Fixed Points of Asymptotically Regular Mappings." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 43, no. 3 (December 1987): 328–46. http://dx.doi.org/10.1017/s1446788700029621.

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Some results on fixed points of asymptotically regular mappings are obtained in complete metric spaces and normed linear spaces.The structure of the set of common fixed points is also discussed in Banach spaces. Our work generalizes essentially known results of Das and Naik, Fisher, Jaggi, Jungck, Rhoades, Singh and Tiwari and several others.

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21

Basso, Giuliano, and Benjamin Miesch. "Conical geodesic bicombings on subsets of normed vector spaces." Advances in Geometry 19, no. 2 (April 24, 2019): 151–64. http://dx.doi.org/10.1515/advgeom-2018-0008.

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Abstract We establish existence and uniqueness results for conical geodesic bicombings on subsets of normed vector spaces. Concerning existence, we give a first example of a convex geodesic bicombing that is not consistent. Furthermore, we show that under a mild geometric assumption on the norm a conical geodesic bicombing on an open subset of a normed vector space locally consists of linear geodesics. As an application, we obtain by the use of a Cartan–Hadamard type result that if a closed convex subset of a Banach space has non-empty interior, then it admits a unique consistent conical geodesic bicombing, namely the one given by the linear segments.

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22

Gordon, Y., O. Guédon, M. Meyer, and A. Pajor. "Random Euclidean sections of some classical Banach spaces." MATHEMATICA SCANDINAVICA 91, no. 2 (December 1, 2002): 247. http://dx.doi.org/10.7146/math.scand.a-14389.

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Using probabilistic arguments, we give precise estimates of the Banach-Mazur distance of subspaces of the classical $\ell_q^n$ spaces and of Schatten classes of operators $S_q^n$ for $q \ge 2$ to the Euclidean space. We also estimate volume ratios of random subspaces of a normed space with respect to subspaces of quotients of $\ell_q$. Finally, the preceeding methods are applied to give estimates of Gelfand numbers of some linear operators.

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23

Eyland, Roger, and Bernice Sharp. "A factor theorem for locally convex differentiability spaces." Bulletin of the Australian Mathematical Society 43, no. 1 (February 1991): 101–13. http://dx.doi.org/10.1017/s0004972700028811.

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The main result of this paper is that a continuous convex function with domain in a locally convex space factors through a normed space. In a recent paper by Sharp, topological linear spaces are categorised according to the differentiability properties of their continuous convex functions; we show that under suitable conditions the classification is preserved by linear maps. A technique for deducing results for locally convex spaces from Banach space theory is an immediate consequence. Examples are given and Asplund C(S) spaces are characterised.

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24

Martini, Horst, and Senlin Wu. "Concurrent and parallel chords of spheres in normed linear spaces." Studia Scientiarum Mathematicarum Hungarica 47, no. 4 (December 1, 2010): 505–12. http://dx.doi.org/10.1556/sscmath.2009.1147.

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We give a geometric characterization of inner product spaces among all finite dimensional real Banach spaces via concurrent chords of their spheres. Namely, let x be an arbitrary interior point of a ball of a finite dimensional normed linear space X. If the locus of the midpoints of all chords of that ball passing through x is a homothetical copy of the unit sphere of X, then the space X is Euclidean. Two further characterizations of the Euclidean case are given by considering parallel chords of 2-sections through the midpoints of balls.

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25

Pasupathi, Agilan, Julietraja Konsalraj, Nahid Fatima, Vallinayagam Velusamy, Nabil Mlaiki, and Nizar Souayah. "Direct and Fixed-Point Stability–Instability of Additive Functional Equation in Banach and Quasi-Beta Normed Spaces." Symmetry 14, no. 8 (August 16, 2022): 1700. http://dx.doi.org/10.3390/sym14081700.

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Over the last few decades, a certain interesting class of functional equations were developed while obtaining the generating functions of many system distributions. This class of equations has numerous applications in many modern disciplines such as wireless networks and communications. The Ulam stability theorem can be applied to numerous functional equations in investigating the stability when approximated in Banach spaces, Banach algebra, and so on. The main focus of this study is to analyse the relationship between functional equations, Hyers–Ulam–Rassias stability, Banach space, quasi-beta normed spaces, and fixed-point theory in depth. The significance of this work is the incorporation of the stability of the generalised additive functional equation in Banach space and quasi-beta normed spaces by employing concrete techniques like direct and fixed-point theory methods. They are powerful tools for narrowing down the mathematical models that describe a wide range of events. Some classes of functional equations, in particular, have lately emerged from a variety of applications, such as Fourier transforms and the Laplace transforms. This study uses linear transformation to explain our functional equations while providing suitable examples.

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26

Harikrishnan, P. K., Bernardo La Fuerza Guillén, and K. T. Ravindran. "Accretive operators and Banach Alaoglu theorem in Linear 2-normed spaces." Proyecciones (Antofagasta) 30, no. 3 (December 2011): 319–27. http://dx.doi.org/10.4067/s0716-09172011000300004.

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27

Park, Choonkil, and Themistocles M. Rassias. "Inequalities in Additive N-isometries on Linear N-normed Banach Spaces." Journal of Inequalities and Applications 2007 (2007): 1–13. http://dx.doi.org/10.1155/2007/70597.

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28

GHEORGHE, DANA. "A KATO PERTURBATION-TYPE RESULT FOR OPEN LINEAR RELATIONS IN NORMED SPACES." Bulletin of the Australian Mathematical Society 79, no. 1 (February 2009): 85–101. http://dx.doi.org/10.1017/s0004972708001056.

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AbstractUsing some techniques of perturbation theory for Banach space complexes, we obtain necessary and sufficient conditions for the stability of the topological index of an open linear relation under small (with respect to the gap topology) perturbations with linear relations.

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29

Navascués, María A., Sangita Jha, Arya K. B. Chand, and Ram N. Mohapatra. "Iterative Schemes Involving Several Mutual Contractions." Mathematics 11, no. 9 (April 24, 2023): 2019. http://dx.doi.org/10.3390/math11092019.

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In this paper, we introduce the new concept of mutual Reich contraction that involves a pair of operators acting on a distance space. We chose the framework of strong b-metric spaces (generalizing the standard metric spaces) in order to add a more extended underlying structure. We provide sufficient conditions for two mutually Reich contractive maps in order to have a common fixed point. The result is extended to a family of operators of any cardinality. The dynamics of iterative discrete systems involving this type of self-maps is studied. In the case of normed spaces, we establish some relations between mutual Reich contractivity and classical contractivity for linear operators. Then, we introduce the new concept of mutual functional contractivity that generalizes the concept of classical Banach contraction, and perform a similar study to the Reich case. We also establish some relations between mutual functional contractions and Banach contractivity in the framework of quasinormed spaces and linear mappings. Lastly, we apply the obtained results to convolutional operators that had been defined by the first author acting on Bochner spaces of integrable Banach-valued curves.

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Nădăban, Sorin. "Fuzzy Continuous Mappings on Fuzzy F-spaces." Mathematics 10, no. 20 (October 12, 2022): 3746. http://dx.doi.org/10.3390/math10203746.

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In the present paper, we first introduce different types of fuzzy continuity for mappings between fuzzy F-normed linear spaces and the relations between them are investigated. Secondly, the principles of fuzzy functional analysis are established in the context of fuzzy F-spaces. More precisely, based on the fact that fuzzy continuity and topological continuity are equivalent, we obtain the closed graph theorem and the open mapping theorem. Using Zabreiko’s lemma, we prove the uniform bounded principle and Banach–Steinhaus theorem. Finally, some future research directions are presented.

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31

Gabeleh, Moosa, S. I. Ezhil Manna, A. Anthony Eldred, and Olivier Olela Otafudu. "Strong and weak convergence of Ishikawa iterations for best proximity pairs." Open Mathematics 18, no. 1 (February 19, 2020): 10–21. http://dx.doi.org/10.1515/math-2020-0002.

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Abstract Let A and B be nonempty subsets of a normed linear space X. A mapping T : A ∪ B → A ∪ B is said to be a noncyclic relatively nonexpansive mapping if T(A) ⊆ A, T(B) ⊆ B and ∥Tx − Ty∥ ≤ ∥x − y∥ for all (x, y) ∈ A × B. A best proximity pair for such a mapping T is a point (p, q) ∈ A × B such that p = Tp, q = Tq and d(p, q) = dist(A, B). In this work, we introduce a geometric notion of proximal Opiaľs condition on a nonempty, closed and convex pair of subsets of strictly convex Banach spaces. By using this geometric notion, we study the strong and weak convergence of the Ishikawa iterative scheme for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. We also establish a best proximity pair theorem for noncyclic contraction type mappings in the setting of strictly convex Banach spaces.

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32

Konwar, Nabanita, and Pradip Debnath. "Continuity and Banach contraction principle in intuitionistic fuzzy n-normed linear spaces." Journal of Intelligent & Fuzzy Systems 33, no. 4 (September 22, 2017): 2363–73. http://dx.doi.org/10.3233/jifs-17500.

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33

Alikhani, Morteza. "Some applications of p-(DPL) sets." Filomat 37, no. 5 (2023): 1367–76. http://dx.doi.org/10.2298/fil2305367a.

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In this paper, we introduce a new class of subsets of class bounded linear operators between Banach spaces which is called p-(DPL) sets. Then, the relationship between these sets with equicompact sets is investigated. Moreover, we define p-version of Right sequentially continuous differentiable mappings and get some characterizations of these mappings. Finally, we prove that a mapping f : X ? Y between real Banach spaces is Fr?chet differentiable and f? takes bounded sets into p-(DPL) sets if and only if f may be written in the form f = 1?S where the intermediate space is normed, S is a Dunford-Pettis p-convergent operator, and g is a G?teaux differentiable mapping with some additional properties.

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García-Pacheco, Francisco Javier, and Francisco Javier Pérez-Fernández. "Pre-Schauder Bases in Topological Vector Spaces." Symmetry 11, no. 8 (August 9, 2019): 1026. http://dx.doi.org/10.3390/sym11081026.

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A Schauder basis in a real or complex Banach space X is a sequence ( e n ) n ∈ N in X such that for every x ∈ X there exists a unique sequence of scalars ( λ n ) n ∈ N satisfying that x = ∑ n = 1 ∞ λ n e n . Schauder bases were first introduced in the setting of real or complex Banach spaces but they have been transported to the scope of real or complex Hausdorff locally convex topological vector spaces. In this manuscript, we extend them to the setting of topological vector spaces over an absolutely valued division ring by redefining them as pre-Schauder bases. We first prove that, if a topological vector space admits a pre-Schauder basis, then the linear span of the basis is Hausdorff and the series linear span of the basis minus the linear span contains the intersection of all neighborhoods of 0. As a consequence, we conclude that the coefficient functionals are continuous if and only if the canonical projections are also continuous (this is a trivial fact in normed spaces but not in topological vector spaces). We also prove that, if a Hausdorff topological vector space admits a pre-Schauder basis and is w * -strongly torsionless, then the biorthogonal system formed by the basis and its coefficient functionals is total. Finally, we focus on Schauder bases on Banach spaces proving that every Banach space with a normalized Schauder basis admits an equivalent norm closer to the original norm than the typical bimonotone renorming and that still makes the basis binormalized and monotone. We also construct an increasing family of left-comparable norms making the normalized Schauder basis binormalized and show that the limit of this family is a right-comparable norm that also makes the normalized Schauder basis binormalized.

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35

Gishlarkaev, Vakha I. "Fourier transform method for partial differential equations. Part 2. Existence and uniqueness of solutions to the Cauchy problem for linear equations." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 10, no. 1 (2023): 21–35. http://dx.doi.org/10.21638/spbu01.2023.103.

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The article proposes a method for analyzing the Cauchy problem for a wide class of evolutionary linear partial differential equations with variable coefficients. By applying the (inverse) Fourier transform, the original equation is reduced to an integro-differential equation, which can be considered as an ordinary differential equation in the corresponding Banach space. The selection of this space is carried out in such a way that the principle of contraction mappings can be used. To carry out the corresponding estimates for the operators generated by the transformed equation, we impose the conditions of finiteness in the space variable for the inverse Fourier transform of the coefficients, and the spaces of the coefficients of the original equation are determined from the Paley-Wiener Fourier transform theorems. In this case, the apparatus of the theory of the Bochner integral in pseudo-normed spaces, countably-normed spaces and Sobolev spaces is used. Classes of functions are distinguished in which the existence and uniqueness of solutions are proved. For equations with coefficients with separated variables, exact solutions are obtained in the form of a Fourier transform of finite sums for operator exponentials.

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36

Solatikia, Farnaz, Erdem Kiliç, and Gerhard Wilhelm Weber. "Fuzzy optimization for portfolio selection based on Embedding Theorem in Fuzzy Normed Linear Spaces." Organizacija 47, no. 2 (May 1, 2014): 90–97. http://dx.doi.org/10.2478/orga-2014-0010.

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Abstract Background: This paper generalizes the results of Embedding problem of Fuzzy Number Space and its extension into a Fuzzy Banach Space C(Ω) × C(Ω), where C(Ω) is the set of all real-valued continuous functions on an open set Ω. Objectives: The main idea behind our approach consists of taking advantage of interplays between fuzzy normed spaces and normed spaces in a way to get an equivalent stochastic program. This helps avoiding pitfalls due to severe oversimplification of the reality. Method: The embedding theorem shows that the set of all fuzzy numbers can be embedded into a Fuzzy Banach space. Inspired by this embedding theorem, we propose a solution concept of fuzzy optimization problem which is obtained by applying the embedding function to the original fuzzy optimization problem. Results: The proposed method is used to extend the classical Mean-Variance portfolio selection model into Mean Variance-Skewness model in fuzzy environment under the criteria on short and long term returns, liquidity and dividends. Conclusion: A fuzzy optimization problem can be transformed into a multiobjective optimization problem which can be solved by using interactive fuzzy decision making procedure. Investor preferences determine the optimal multiobjective solution according to alternative scenarios.

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37

Li, Ruini, and Jianrong Wu. "Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces." AIMS Mathematics 7, no. 3 (2022): 3290–302. http://dx.doi.org/10.3934/math.2022183.

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<abstract> <p>In this paper, we first study continuous linear functionals on a fuzzy quasi-normed space, obtain a characterization of continuous linear functionals, and point out that the set of all continuous linear functionals forms a convex cone and can be equipped with a weak fuzzy quasi-norm. Next, we prove a theorem of Hahn-Banach type and two separation theorems for convex subsets of fuzzy quasinormed spaces.</p> </abstract>

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38

Jiménez-Vargas, A., Miguel Lacruz, and Moisés Villegas-Vallecillos. "Essential Norm of Composition Operators on Banach Spaces of Hölder Functions." Abstract and Applied Analysis 2011 (2011): 1–13. http://dx.doi.org/10.1155/2011/590853.

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Let(X,d)be a pointed compact metric space, let0<α<1, and letφ:X→Xbe a base point preserving Lipschitz map. We prove that the essential norm of the composition operatorCφinduced by the symbolφon the spaceslip0(X,dα)andLip0(X,dα)is given by the formula‖Cφ‖e=limt→0 sup⁡0<d(x, y)<t(d(φ(x),φ(y))α/d(x,y)α)whenever the dual spacelip0(X,dα)∗has the approximation property. This happens in particular whenXis an infinite compact subset of a finite-dimensional normed linear space.

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39

Bonales, Fernando Garibay, and Rigoberto Vera Mendoza. "A formula to calculate the spectral radius of a compact linear operator." International Journal of Mathematics and Mathematical Sciences 20, no. 3 (1997): 585–88. http://dx.doi.org/10.1155/s0161171297000793.

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There is a formula (Gelfand's formula) to find the spectral radius of a linear operator defined on a Banach space. That formula does not apply even in normed spaces which are not complete. In this paper we show a formula to find the spectral radius of any linear and compact operatorTdefined on a complete topological vector space, locally convex. We also show an easy way to find a non-trivialT-invariant closed subspace in terms of Minkowski functional.

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40

Astaneh, A. A. "Inscribed centers, reflexivity, and some applications." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 41, no. 3 (December 1986): 317–24. http://dx.doi.org/10.1017/s1446788700033759.

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AbstractWe first define an inscribed center of a bounded convex body in a normed linear space as the center of a largest open ball contained in it (when such a ball exists). We then show that completeness is a necessary condition for a normed linear space to admit inscribed centers. We show that every weakly compact convex body in a Banach space has at least one inscribed center, and that admitting inscribed centers is a necessary and sufficient condition for reflexivity. We finally apply the concept of inscribed center to prove a type of fixed point theorem and also deduce a proposition concerning so-called Klee caverns in Hilbert spaces.

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Sidorov, Nikolai A., Aliona I. Dreglea, and Denis N. Sidorov. "Generalisation of the Frobenius Formula in the Theory of Block Operators on Normed Spaces." Mathematics 9, no. 23 (November 28, 2021): 3066. http://dx.doi.org/10.3390/math9233066.

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The efficient construction and employment of block operators are vital for contemporary computing, playing an essential role in various applications. In this paper, we prove a generalisation of the Frobenius formula in the setting of the theory of block operators on normed spaces. A system of linear equations with the block operator acting in Banach spaces is considered. Existence theorems are proved, and asymptotic approximations of solutions in regular and irregular cases are constructed. In the latter case, the solution is constructed in the form of a Laurent series. The theoretical approach is illustrated with an example, the construction of solutions for a block equation leading to a method of solving some linear integrodifferential system.

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42

Nadaban, Sorin. "Fuzzy Continuous Mappings in Fuzzy Normed Linear Spaces." International Journal of Computers Communications & Control 10, no. 6 (October 3, 2015): 74. http://dx.doi.org/10.15837/ijccc.2015.6.2074.

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In this paper we continue the study of fuzzy continuous mappings in fuzzy normed linear spaces initiated by T. Bag and S.K. Samanta, as well as by I. Sadeqi and F.S. Kia, in a more general settings. Firstly, we introduce the notion of uniformly fuzzy continuous mapping and we establish the uniform continuity theorem in fuzzy settings. Furthermore, the concept of fuzzy Lipschitzian mapping is introduced and a fuzzy version for Banach’s contraction principle is obtained. Finally, a special attention is given to various characterizations of fuzzy continuous linear operators. Based on our results, classical principles of functional analysis (such as the uniform boundedness principle, the open mapping theorem and the closed graph theorem) can be extended in a more general fuzzy context.

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Nakasho, Kazuhisa, Yuichi Futa, and Yasunari Shidama. "Topological Properties of Real Normed Space." Formalized Mathematics 22, no. 3 (September 1, 2014): 209–23. http://dx.doi.org/10.2478/forma-2014-0024.

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Summary In this article, we formalize topological properties of real normed spaces. In the first part, open and closed, density, separability and sequence and its convergence are discussed. Then we argue properties of real normed subspace. Then we discuss linear functions between real normed speces. Several kinds of subspaces induced by linear functions such as kernel, image and inverse image are considered here. The fact that Lipschitz continuity operators preserve convergence of sequences is also refered here. Then we argue the condition when real normed subspaces become Banach’s spaces. We also formalize quotient vector space. In the last session, we argue the properties of the closure of real normed space. These formalizations are based on [19](p.3-41), [2] and [34](p.3-67).

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44

GUEYE, M. M., M. SENE, M. NDIAYE, and N. DJITTE. "Explicit algorithms for J-fixed points of some non linear mappings in certain Banach spaces." Creative Mathematics and Informatics 29, no. 1 (January 30, 2020): 27–36. http://dx.doi.org/10.37193/cmi.2020.01.04.

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Let E be a real normed linear space and E∗ its dual. In a recent work, Chidume et al. [Chidume, C. E. and Idu, K. O., Approximation of zeros of bounded maximal monotone mappings, solutions of hammerstein integral equations and convex minimizations problems, Fixed Point Theory and Applications, 97 (2016)] introduced the new concepts of J-fixed points and J-pseudocontractive mappings and they shown that a mapping A : E → 2 E∗ is monotone if and only if the map T := (J −A) : E → 2 E∗ is J-pseudocontractive, where J is the normalized duality mapping of E. It is our purpose in this work to introduce an algorithm for approximating J-fixed points of J-pseudocontractive mappings. Our results are applied to approximate zeros of monotone mappings in certain Banach spaces. The results obtained here, extend and unify some recent results in this direction for the class of maximal monotone mappings in uniformly smooth and strictly convex real Banach spaces. Our proof is of independent interest.

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Sharifi, A. R., Azadi Kenary, B. Yousefi, and R. Soltani. "HUR-approximation of an ELTA functional equation." Filomat 34, no. 13 (2020): 4311–28. http://dx.doi.org/10.2298/fil2013311s.

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The main goal of this paper is study of the Hyers-Ulam-Rassias stability (briefly HUR-approximation) of the following Euler-Lagrange type additive(briefly ELTA) functional equation ?nj=1f (1/2 ?1?i?n,i?j rixi- 1/2 rjxj) + ?ni=1 rif(xi)=nf (1/2 ?ni=1 rixi) where r1,..., rn ? R, ?ni=k rk?0, and ri,rj?0 for some 1? i < j ? n, in fuzzy normed spaces. The concept of HUR-approximation originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

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46

Bînzar, Tudor, Flavius Pater, and Sorin Nădăban. "Fixed-Point Theorems in Fuzzy Normed Linear Spaces for Contractive Mappings with Applications to Dynamic-Programming." Symmetry 14, no. 10 (September 20, 2022): 1966. http://dx.doi.org/10.3390/sym14101966.

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The aim of this paper is to provide new ways of dealing with dynamic programming using a context of newly proven results about fixed-point problems in linear spaces endowed with a fuzzy norm. In our paper, the general framework is set to fuzzy normed linear spaces as they are defined by Nădăban and Dzitac. When completeness is required, we will use the George and Veeramani (G-V) setup, which, for our purposes, we consider to be more suitable than Grabiec-completeness. As an important result of our work, we give an original proof for a version of Banach’s fixed-point principle on this particular setup of fuzzy normed spaces, a variant of Jungck’s fixed-point theorem in the same setup, and they are proved in G-V-complete fuzzy normed spaces, paving the way for future developments in various fields within this framework, where our application of dynamic programming makes a proper example. As the uniqueness of almost every dynamic programming problem is necessary, the fixed-point theorems represent an important tool in achieving that goal.

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Erdoğan, Ezgi, and Enrique A. Sánchez Pérez. "Approximate Diagonal Integral Representations and Eigenmeasures for Lipschitz Operators on Banach Spaces." Mathematics 10, no. 2 (January 12, 2022): 220. http://dx.doi.org/10.3390/math10020220.

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A new stochastic approach for the approximation of (nonlinear) Lipschitz operators in normed spaces by their eigenvectors is shown. Different ways of providing integral representations for these approximations are proposed, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex. We use the recently introduced notion of eigenmeasure and focus attention on procedures for extending a function for which the eigenvectors are known, to the whole space. We provide information on natural error bounds, thus giving some tools to measure to what extent the map can be considered diagonal with few errors. In particular, we show an approximate spectral theorem for Lipschitz operators that verify certain convexity properties.

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Bahyrycz, Anna, and Justyna Sikorska. "On Stability of a General n-Linear Functional Equation." Symmetry 15, no. 1 (December 21, 2022): 19. http://dx.doi.org/10.3390/sym15010019.

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Let X be a linear space over K∈{R,C}, Y be a real or complex Banach space and f:Xn→Y. With some fixed aji,Ci1…in∈K (j∈{1,…,n}, i,ik∈{1,2}, k∈{1,…,n}), we study, using the direct and the fixed point methods, the stability and the general stability of the equation f(a11x11+a12x12,…,an1xn1+an2xn2)=∑1≤i1,…,in≤2Ci1…inf(x1i1,…,xnin), for all xjij∈X (j∈{1,…,n},ij∈{1,2}). Our paper generalizes several known results, among others concerning equations with symmetric coefficients, such as the multi-Cauchy equation or the multi-Jensen equation as well as the multi-Cauchy–Jensen equation. We also prove the hyperstability of the above equation in m-normed spaces with m≥2.

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49

Iovino, José. "On the maximality of logics with approximations." Journal of Symbolic Logic 66, no. 4 (December 2001): 1909–18. http://dx.doi.org/10.2307/2694984.

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In this paper we analyze some aspects of the question of using methods from model theory to study structures of functional analysis.By a well known result of P. Lindström, one cannot extend the expressive power of first order logic and yet preserve its most outstanding model theoretic characteristics (e.g., compactness and the Löwenheim-Skolem theorem). However, one may consider extending the scope of first order in a different sense, specifically, by expanding the class of structures that are regarded as models (e.g., including Banach algebras or other structures of functional analysis), and ask whether the resulting extensions of first order model theory preserve some of its desirable characteristics.A formal framework for the study of structures based on Banach spaces from the perspective of model theory was first introduced by C. W. Henson in [8] and [6]. Notions of syntax and semantics for these structures were defined, and it was shown that using them one obtains a model theoretic apparatus that satisfies many of the fundamental properties of first order model theory. For instance, one has compactness, Löwenheim-Skolem, and omitting types theorems. Further aspects of the theory, namely, the fundamentals of stability and forking, were first introduced in [10] and [9].The classes of mathematical structures formally encompassed by this framework are normed linear spaces, possibly expanded with additional structure, e.g., operations, real-valued relations, and constants. This notion subsumes wide classes of structures from functional analysis. However, the restriction that the universe of a structure be a normed space is not necessary. (This restriction has a historical, rather than technical origin; specifically, the development of the theory was originally motivated by questions in Banach space geometry.) Analogous techniques can be applied if the universe is a metric space. Now, when the underlying metric topology is discrete, the resulting model theory coincides with first order model theory, so this logic extends first order in the sense described above. Furthermore, without any cost in the mathematical complexity, one can also work in multi-sorted contexts, so, for instance, one sort could be an operator algebra while another is. say, a metric space.

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50

Garcia-Pacheco, Francisco Javier. "Regularity in Topological Modules." Mathematics 8, no. 9 (September 13, 2020): 1580. http://dx.doi.org/10.3390/math8091580.

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The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules is a relatively new trend that has enriched the literature of Functional Analysis with deeper classical results as well as with pathological phenomena. Following this trend, it has been recently proved that every real or complex Hausdorff locally convex topological vector space with dimension greater than or equal to 2 has a balanced and absorbing subset with empty interior. Here we propose an extension of this result to topological modules over topological rings. A sufficient condition is provided to accomplish this extension. This sufficient condition is a new property in topological module theory called strong open property. On the other hand, topological regularity of closed balls and open balls in real or complex normed spaces is a trivial fact. Sufficient conditions, related to the strong open property, are provided on seminormed modules over an absolutely semivalued ring for closed balls to be regular closed and open balls to be regular open. These sufficient conditions are in fact characterizations when the seminormed module is the absolutely semivalued ring. These characterizations allow the provision of more examples of closed-unit neighborhoods of zero. Consequently, the closed-unit ball of any unital real Banach algebra is proved to be a closed-unit zero-neighborhood. We finally transport all these results to topological modules over topological rings to obtain nontrivial regular closed and regular open neighborhoods of zero. In particular, if M is a topological R-module and m∗∈M∗ is a continuous linear functional on M which is open as a map between topological spaces, then m∗−1(int(B)) is regular open and m∗−1(B) is regular closed, for B any closed-unit zero-neighborhood in R.

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