Abstract
A well known result of R. R. Phelps (1960) asserts that in order that every linear continuous functional, defined on a subspace \(Y\) of a real normed space \(X\), have a unique norm preserving extension it is necessary and sufficient that its annihilator \(Y^{\perp}\) be a Chebyshevian subspace of \(X\ast\). The aim of this note is to show that this result holds also in the case of spaces with asymmetric norm.
Authors
Costica Mustăţa
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academ, Romania
Keywords
Asymmetric normed spaces; extensions preserving asymmetricnorm; best approximation
Paper coordinates
C. Mustăţa, On the uniqueness of extension and unique best approximation in the dual of an asymmetric normed linear space, Rev. Anal. Numer. Theor. Approx. 32 (2003) no. 2, 187-192.
About this paper
Journal
Revue d’Analyse Numerique et de Theorie de l’Approximation
Publisher Name
Publising Romanian Academy
DOI
Print ISSN
2457-6794
Online ISSN
2501-059X
google scholar link
[2] Borodin, P. A.,The Banach–Mazur Theorem for spaces with asymmetric norm andits applications in convex analysis, Mathematical Notes,69, no. 3, pp. 298–305, 2001.
[3] Cobzas S.,Phelps type duality results in best approximation,Rev. Anal. Numer.Theor. Approx.,31, no. 1, pp. 29–43, 2002.
[4] Dolzhenko, E. P.andSevast’yanov, E. A.,Approximation with sign-sensitiveweights, Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 62, no. 6,pp. 59–102, 1998 and63, no. 3, pp. 77–118, 1999.
[5] Ferrer, J., Gregori, V.andAlegre, C.,Quasi-uniform structures in linear lattices,Rocky Mountain J. Math.,23, pp. 877–884, 1993.
[6] Garcia-Raffi, L. M., Romaguera, S.and Sanchez Perez, E. A.,Extension ofasymmetric norms to linear spaces, Rend. Istit. Mat. Trieste,XXXIII, pp. 113–125,2001.
[7] Garcia-Raffi, L. M., Romaguera S.and Sanchez-Perez, E. A.,The dual space ofan asymmetric normed linear space,Quaestiones Mathematicae,26, pp. 83–96, 2003.
[8] Krein, M. G.and Nudel’man, A. A.,The Markov Moment Problem and ExtremumProblems, Nauka, Moscow, 1973 (in Russian).
[9] Kopperman, R. D.,All topologies come from generalized metrics, Amer. Math.Monthly,95, pp. 89–97, 1988.
[10] McShane, E. J.,Extension of range of functions, Bull. Amer. Math. Soc.,40, pp. 847–842, 1934.
[11] Mustata, C.,Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev.Anal. Numer. Theor. Approx.,30, no. 1, pp. 61–67, 2001.
[12] Mustata, C.,Extensions of convex semi-Lipschitz functions on quasi-metric linearspaces, Seminaire de la Theorie de la Meileure Approximation Convexite et Optimiza-tion, Cluj-Napoca, le 29 novembre, pp. 85–92, 2001.
[13] Mustata, C.,A Phelps type theorem for spaces with asymmetric norms, Bul. Stiint.Univ. Baia-Mare, Ser. B,XVIII, no. 2, pp. 275–280, 2002.
[14] Phelps, R. R.,Uniqueness of Hahn–Banach extension and unique best approximation,Trans. Amer. Math. Soc.,95, pp. 238–255, 1960.
[15] Romaguera, S.and Sanchis, M.,Semi-Lipschitz functions and best approximation inquasi-metric spaces, J. Approx. Theory,103, pp. 292–301, 2000.
[16] Singer, I.,Best approximation in normed linear spaces by elements of linear subspaces,Ed. Acad. RSR, Bucharest, 1967 (in Romanian); English Translation, Springer, Berlin,1970