On the uniqueness of extension and unique best approximation in the dual of an asymmetric normed linear space

Costică Mustăţa

doi:10.33993/jnaat322-747

Authors

Costică Mustăţa Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy, Romania

DOI:

https://doi.org/10.33993/jnaat322-747

Keywords:

asymmetric normed spaces, extensions preserving asymmetric norm, best approximation

Abstract views: 354

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 \(Y^\bot\) 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.

Downloads

Download data is not yet available.

References

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. DOI: 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. DOI: 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. DOI: https://doi.org/10.1216/rmjm/1181072529

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. DOI: 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. DOI: 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. DOI: 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. DOI: https://doi.org/10.1090/S0002-9947-1960-0113125-4

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. DOI: 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. DOI: https://doi.org/10.1007/978-3-662-41583-2