## Abstract

In this paper we shall present some results on spaces with asymmetric seminorms, with emphasis on best approximation problems in such spaces.

## Authors

**Stefan Cobzas**

Babes-Bolyai University, Cluj-Napoca, Romania

**Costică Mustăţa**

“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

## Keywords

Spaces with asymmetric norm; best approximation; Hahn-Banach theorem; characterization of best approximation.

## Paper coordinates

S. Cobzas, C. Mustăţa, *Best approximation in spaces with asymmetric norm*, Rev. Anal. Numer. Theor. Approx. 35 (2006) no. 1, 17-31.

## About this paper

##### Journal

Revue d’Analyse Numer. Theor. Approx.

##### Publisher Name

Publishing House of the Romanian Academy

##### Print ISSN

2501-059X

##### Online ISSN

2457-6794

google scholar link

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[7] Borodin, P. A.,The Banach-Mazur theorem for spaces with an asymmetric norm andits applications in convex analysis, Mat. Zametki,69, no. 3, pp. 329–337, 2001.

[8] Cobzas, S.,Phelps type duality results in best approximation, Rev. Anal. Numer.Theor. Approx.,31, no. 1, pp. 29–43, 2002.

[9]___,Separation of convex sets and best approximation in spaces with asymmetricnorm, Quaest. Math.,27, no. 3, 275–296, 2004.

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[11] Cobzas, S. and Mustata, C.,Extension of bilinear functionals and best approximationin 2-normed spaces, Studia Univ. Babes-Bolyai, Mathematica, 43, pp. 1–13, 1998.

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[13] De Blasi, F. S. and Myjak, J., On a generalized best approximation problem, J. Approx.Theory,94, no. 1, pp. 54–72, 1998.

[14] Dolzhenko, E. P. andSevastyanov, E. A.,Approximations with a sign-sensitiveweight (existence and uniqueness theorems), Izv. Ross. Akad. Nauk Ser. Mat.,62, no. 6,pp. 59–102,1998.

[15]___,Sign-sensitive approximations, J. Math. Sci. (New York),91, no. 5, pp. 3205–3257, 1998.

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[25] Chong Liand Renxing Ni, Derivatives of generalized distance functions and existenceof generalized nearest points, J. Approx. Theory,115, no. 1, pp. 44–55, 2002.

[26] Mohebi, H.,On quasi-Chebyshev subspaces of Banach spaces, J. Approx. Theory,107,no. 1, pp. 87–95, 2000.

[27]___,Pseudo-Chebyshev subspaces inL1, Korean J. Comput. Appl. Math.,7, no. 2,pp. 465–475, 2000.

[28]___,On pseudo-Chebyshev subspaces in normed linear spaces, Math. Sci. Res. Hot-Line,5, no. 9, pp. 29–45, 2001.

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[33]___,On the extremal semi-Lipschitz functions, Rev. Anal. Numer. Theor. Approx.,31, no. 1, pp. 103–108, 2002.

[34]___,A Phelps type theorem for spaces with asymmetric norms, Bul. Stiint. Univ.Baia Mare, Ser. B, Matematica-Informatica,18, no. 2, pp. 275–280, 2002.

[35]___,On the uniqueness of the extension and unique best approximation in the dual ofan asymmetric linear space, Rev. Anal. Numer. Theor. Approx.,32, no. 2, pp. 187–192,2003.

[2]___,On the Hahn-Banach theorem in certain linear quasi-uniform structures, ActaMath. Hungar.,82, no. 4, pp. 325–330, 1999.

[3] Alimov, A. R.,The Banach-Mazur theorem for spaces with nonsymmetric distance,Uspekhi Mat. Nauk, 58, no. 2, pp. 159–160, 2003.

[4] Babenko, V. F.,Nonsymmetric approximations in spaces of summable functions,Ukrain. Mat. Zh., 34, no. 4, pp. 409–416, 538, 1982.

[5]___,Nonsymmetric extremal problems of approximation theory, Dokl. Akad. NaukSSSR,269, no. 3, pp. 521–524, 1983.

[6]___,Duality theorems for certain problems of the theory of approximation, Currentproblems in real and complex analysis, Akad. Nauk Ukrain. SSR Inst. Mat., Kiev,pp. 3–13, 148, 1984.

[7] Borodin, P. A.,The Banach-Mazur theorem for spaces with an asymmetric norm andits applications in convex analysis, Mat. Zametki,69, no. 3, pp. 329–337, 2001.

[8] Cobzas, S.,Phelps type duality results in best approximation, Rev. Anal. Numer.Theor. Approx.,31, no. 1, pp. 29–43, 2002.

[9]___,Separation of convex sets and best approximation in spaces with asymmetricnorm, Quaest. Math.,27, no. 3, 275–296, 2004.

[10]___,Asymmetric locally convex spaces,Int. J. Math. Math. Sci., no. 16, 2585–2608,2005.

[11] Cobzas, S. and Mustata, C.,Extension of bilinear functionals and best approximationin 2-normed spaces, Studia Univ. Babes-Bolyai, Mathematica, 43, pp. 1–13, 1998.

[12]___,Extension of bounded linear functionals and best approximation in spaces withasymmetric norm, Rev. Anal. Numer. Theor. Approx.,33, no. 1, pp. 39–50, 2004.

[13] De Blasi, F. S. and Myjak, J., On a generalized best approximation problem, J. Approx.Theory,94, no. 1, pp. 54–72, 1998.

[14] Dolzhenko, E. P. andSevastyanov, E. A.,Approximations with a sign-sensitiveweight (existence and uniqueness theorems), Izv. Ross. Akad. Nauk Ser. Mat.,62, no. 6,pp. 59–102,1998.

[15]___,Sign-sensitive approximations, J. Math. Sci. (New York),91, no. 5, pp. 3205–3257, 1998.

[16]___,Approximation with a sign-sensitive weight (stability, applications to snake the-ory and Hausdorff approximations), Izv. Ross. Akad. Nauk Ser. Mat.,63, no. 3, pp.77–118, 1999.

[17] Ferrer, J.,Gregori, V. andAlegre, C.,Quasi-uniform structures in linear lattices,Rocky Mountain J. Math.,23, no. 3, pp. 877–884, 1993.

[18] Garcıa-Raffi, L. M.,Romaguera, S. and Sanchez Perez, E. A.,Extensions of asymmetric norms to linear spaces, Rend. Istit. Mat. Univ. Trieste,33, nos. 1–2, 113–125, 2001.

[19]___,The bicompletion of an asymmetric normed linear space, Acta Math. Hungar.,97, no. 3, pp. 183–191, 2002.

[20]___,Sequence spaces and asymmetric norms in the theory of computational com-plexity, Math. Comput. Modelling,36, nos. 1–2, pp. 1–11, 2002.

[21]___,The dual space of an asymmetric normed linear space, Quaest. Math.,26, no. 1,pp. 83–96, 2003.

[22]___,On Hausdorff asymmetric normed linear spaces, Houston J. Math.,29, no. 3,pp. 717–728 (electronic), 2003.

[23] Krein, M. G. and Nudelman, A. A., The Markov Moment Problem and ExtremumProblems, Nauka, Moscow, 1973 (in Russian), English translation: American Mathe-matical Society, Providence, R.I., 1977.

[24] Chong Li, On wel l posed generalized best approximation problems, J. Approx. Theory,107, no. 1, pp. 96–108, 2000.

[25] Chong Liand Renxing Ni, Derivatives of generalized distance functions and existenceof generalized nearest points, J. Approx. Theory,115, no. 1, pp. 44–55, 2002.

[26] Mohebi, H.,On quasi-Chebyshev subspaces of Banach spaces, J. Approx. Theory,107,no. 1, pp. 87–95, 2000.

[27]___,Pseudo-Chebyshev subspaces inL1, Korean J. Comput. Appl. Math.,7, no. 2,pp. 465–475, 2000.

[28]___,On pseudo-Chebyshev subspaces in normed linear spaces, Math. Sci. Res. Hot-Line,5, no. 9, pp. 29–45, 2001.

[29]___,Quasi-Chebyshev subspaces in dual spaces, J. Nat. Geom.,20, nos. 1-2, pp.33–44, 2001.

[30]___,On pseudo-Chebyshev subspaces in normed linear spaces, J. Nat. Geom.,24,nos. 1–2, pp. 37–56, 2003.

[31] Mohebi, H. andRezapour,Sh.,On weak compactness of the set of extensions of acontinuous linear functional, J. Nat. Geom.,22, nos. 1–2, pp. 91–102, 2002.

[32] Mustata, C.,Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev.Anal. Numer. Theor. Approx.,30, no. 1, pp. 61–67, 2001.

[33]___,On the extremal semi-Lipschitz functions, Rev. Anal. Numer. Theor. Approx.,31, no. 1, pp. 103–108, 2002.

[34]___,A Phelps type theorem for spaces with asymmetric norms, Bul. Stiint. Univ.Baia Mare, Ser. B, Matematica-Informatica,18, no. 2, pp. 275–280, 2002.

[35]___,On the uniqueness of the extension and unique best approximation in the dual ofan asymmetric linear space, Rev. Anal. Numer. Theor. Approx.,32, no. 2, pp. 187–192,2003.

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[37] Phelps, R. R.,Uniqueness of Hahn-Banach extensions and best approximations, Trans.Amer. Marth Soc.,95, pp. 238–255, 1960.

[38] Rezapour, Sh.,∈-weakly Chebyshev subspaces of Banach spaces, Anal. Theory Appl.,19, no. 2, pp. 130–135, 2003.

[39] Romaguera, S. and Sanchis, M.,Semi-Lipschitz functions and best approximation inquasi-metric spaces, J. Approx. Theory,103, no. 2, pp. 292–301, 2000.

[40] Romaguera, S. and Schellekens, M.,Duality and quasi-normability for complexityspaces, Appl. Gen. Topol.,3, no. 1, pp. 91–112, 2002.

[41] Simonov, B. V.,On the element of best approximation in spaces with nonsymmetricquasinorm, Mat. Zametki,74, no. 6, pp. 902–912, 2003.

[42] Singer, I.,Best Approximation in Normed Linear Spaces by Elements of Linear Sub-spaces, Editura Academiei Romane and Springer-Verlag, Bucharest-New York-Berlin,1970.

[43] Zanco, C. and Zucchi, A.,Moduli of rotundity and smoothness for convex bodies,Bolletino U. M. I., (7),7-B, pp. 833–855, 1993