A Phelps type result for spaces with asymmetric norms


If \((X,||\cdot|)\) is a linear space with asymmetric norm and \(Y\) is a \(X\), for every \(f\in Y_{+}^{\ast}\) (the cone of linear bounded functional on \(Y\)) there exists functional \(F\in Y_{+}^{\ast}\) extending \(f\) and preserving the asymmetric norm of \(f\).The problem of uniqueness of the extension in terms of uniqueness of elements of best of \(F\in X_{+}^{\ast}\) by elements of \(Y_{+}^{\perp}=\{G\in X_{+}^{\ast}:\left. G\right \vert _{Y}-0,F\geq G\}\), is discussed.


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


asymmetric norm; extension and approximation.

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C. Mustăţa, A Phelps type result for spaces with asymmetric norms, Bul. Şt. Univ. Baia Mare, Seria B, Fascicola matematică-informatică, 18 (2002) no. 2, 275-280.


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[1] Borodin, P.A.; The Banach-Mazur Theorem for Spaces with Asymetrie Norm and Its Applications in Convex Analysis, Mathematical Notes vol. 69. Nr. 3 (2001), 298-305
[2] Dolzhenko, E.P. and E.A. Sevastyanov, Approximation with sign-sensitive weights. Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sei. Izv. Marh.] 62 (1998) no. 6, 59-102 and 63 (1999) no. 3 77-48.
[3] Ferrer, J., Gregori, V. and C. Alegre, Quasi-uniform structures in linear lattices, Rocky Mountain J. Math. 23 (1993), 877-884
[4] Garcia -Raffi, L.M.; Romaguera S., and Sanchez Pérez E.A., Extension of Asymmetric Norms to Linear Spaces, Rend. Istit. Mat. Trieste XXXIII, 113-125 (2001)
[5] Krein, M.G. and A.A.Nudel’man, The Markov Moment Problem and Extrémům Problems [in Russian], Nauka, Moscow, 1973.
[6] Kopperman, R.D., All topologies come from generalized metrics, Amer. Math. Monthly 95 (1988), 89-97
[7] McShane, E.J., Extension of Range of Functions, Bull. Amer. Math. Soc. 40 (1934), 847-842
[8] Mustata, C., Extensions of Semi-Lipschitz functions on quasi-Metric spaces, Rev. Anal. Numér. Théor. Approx.. 30 (2001) No.l, 61-67
[9] Mustäfa, C., Extensions of convex Semi-Lipschitz Functions on quasi-metric linear spaces, Séminaire de la Théorie de la Meileure Approximation Convexité et Optimization, Cluj-Napoca, le 29 november 2001, 85-92.
[10] Phelps, R.R., Uniqueness of Hahn -Banach Extension and Unique Best Approximation, Trans. Amer. Math. Soc. 95 (1960), 238-255.
[11] Romaguera, S. and M. Sanchis, Semi-Lipschitz Functions and Best Approximation in quasi-Metric Spaces, J. Approx. Theory 103 (2000), 292-301.

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