On the uniqueness of extension and unique best approximation in the dual of an asymmetric normed linear space
Keywords:asymmetric normed spaces, extensions preserving asymmetric norm, best approximation
AbstractA 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 \(Y^\bot\) 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.
Alegre, C., Ferrer, J. and Gregori, V., On the Hahn--Banach theorem in certain linear quasi-uniform structures, Acta Math. Hungar, 82, pp. 315-320, 1999, https://doi.org/10.1023/A:1006692309917.
Borodin, P. A., The Banach-Mazur Theorem for spaces with asymmetric norm and its applications in convex analysis, Mathematical Notes, 69, no. 3, pp. 298-305, 2001, https://doi.org/10.1023/A:1010271105852.
Cobzaş, S., Phelps type duality results in best approximation, Rev. Anal. Numér. Théor. Approx., 31, no. 1, pp. 29-43, 2002, http://ictp.acad.ro/jnaat/journal/article/view/2002-vol31-no1-art5
Dolzhenko, E. P. and Sevast'yanov, E. A., Approximation with sign-sensitive weights, Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 62, no. 6, pp. 59-102, 1998 and 63, no. 3, pp. 77-118, 1999.
Ferrer, J., Gregori, V. and Alegre, C., Quasi-uniform structures in linear lattices, Rocky Mountain J. Math., 23, pp. 877-884, 1993, https://www.jstor.org/stable/44237743.
Garćia-Raffi, L. M., Romaguera, S. and Sanchez Pérez, E. A., Extension of asymmetric norms to linear spaces, Rend. Istit. Mat. Trieste, XXXIII, pp. 113-125, 2001.
Garćia-Raffi, L. M., Romaguera S. and Sánchez-Pérez, E. A., The dual space of an asymmetric normed linear space, Quaestiones Mathematicae, 26, pp. 83-96, 2003, https://doi.org/10.2989/16073600309486046.
Krein, M. G. and Nudel'man, A. A., The Markov Moment Problem and Extremum Problems, Nauka, Moscow, 1973 (in Russian).
Kopperman, R. D., All topologies come from generalized metrics, Amer. Math. Monthly, 95, pp. 89-97, 1988, https://doi.org/10.1080/00029890.1988.11971974.
McShane, E. J., Extension of range of functions, Bull. Amer. Math. Soc., 40, pp. 837-842, 1934, https://doi.org/10.1090/S0002-9904-1934-05978-0.
Mustăţa, C., Extensions of semi-Lipschitz functions on quasi-metric spaces, Rev. Anal. Numér. Théor. Approx., 30, no. 1, pp. 61-67, 2001, http://ictp.acad.ro/jnaat/journal/article/view/2001-vol30-no1-art8
Mustăţa, 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 novembre, pp. 85-92, 2001.
Mustăţa, C., A Phelps type theorem for spaces with asymmetric norms, Bul. Ştiinţ. Univ. Baia-Mare, Ser. B, XVIII, no. 2, pp. 275-280, 2002.
Phelps, R. R., Uniqueness of Hahn--Banach extension and unique best approximation, Trans. Amer. Math. Soc., 95, pp. 238-255, 1960, https://doi.org/10.2307/1993289.
Romaguera, S. and Sanchis, M., Semi-Lipschitz functions and best approximation in quasi-metric spaces, J. Approx. Theory, 103, pp. 292-301, 2000, https://doi.org/10.1006/jath.1999.3439.
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.
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