Phelps type duality results in best approximation

Ştefan Cobzaş

doi:10.33993/jnaat311-706

Authors

Ştefan Cobzaş "Babeş Bolyai" University, Cluj-Napoca, Romania

DOI:

https://doi.org/10.33993/jnaat311-706

Keywords:

best approximation, Hahn-Banach extension, \(M\)-ideals

Abstract views: 294

Abstract

The aim of the present paper is to show that many Phelps type duality result, relating the extension properties of various classes of functions (continuous, linear continuous, bounded bilinear, Hölder-Lipschitz) with the approximation properties of some annihilating spaces, can be derived in a unitary and simple way from a formula for the distance to the kernel of a linear operator, extending the well-known distance formula to hyperplanes in normed spaces. The case of spaces \(c_0\) and \(l^\infty\) is treated in details.

Downloads

Download data is not yet available.

References

Alfsen, E. M. and Effros, E. G., Structure in real Banach spaces, Ann. Math., 96, Part I: 98-128, Part II: 129-173, 1972, https://doi.org/10.2307/1970895 DOI: https://doi.org/10.2307/1970896

Bhaskara Rao, K. P. S. and Bhaskara Rao, M. Theory of Charges-A Study of Finitely Additive Measures, Academic Press, New York, 1983.

Beg, I. and Iqbal, M., Extension of linear 2-operators, Math. Montesnigri, 2, pp. 1--10, 1993.

Cobzaş, S., Best approximation in spaces of bounded vector-valued sequences, Rev. Anal. Numér. Théor. Approx., 23, pp. 63-69, 1994, http://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art6

Cobzaş, S. and Mustăţa, C., Norm-preserving extensions of convex Lipschitz functions, J. Approx. Theory, 24, pp. 236-244, 1978, https://core.ac.uk/download/pdf/82772797.pdf DOI: https://doi.org/10.1016/0021-9045(78)90028-X

Cobzaş, S. and Mustăţa, C., Extension of bilinear operators and best approximation in 2-normed spaces, Rev. Anal. Numér. Théor. Approx., 25, pp. 45-52, 1996, http://ictp.acad.ro/jnaat/journal/article/view/1996-vol25-nos1-2-art7

Cobzaş, S. and Mustăţa, C., Extension of bilinear functionals and best approximation in 2-normed spaces, Studia Univ. Babeş-Bolyai, Series Math. (in print).

Czipser, J. and Géhér, L., Extension of functions satisfying a Lipschitz condition, Acta. Math. Acad. Sci. Hungar., 6, pp. 213-220, 1955, https://doi.org/10.1007/bf02021278 DOI: https://doi.org/10.1007/BF02021278

Day, M. M., Normed Linear Spaces, 3rd Edition, Springer-Verlag, Berlin, 1973. DOI: https://doi.org/10.1007/978-3-662-09000-8

Deutsch, F., Wu Li and Park, S. H., Tietze extensions and continuous selections for the metric projection, J. Approx. Theory, 63, pp. 55-68, 1991, https://doi.org/10.1016/0021-9045(91)90086-P DOI: https://doi.org/10.1016/0021-9045(91)90086-P

Deutsch, F., Wu Li and Mabizela, S., Helly extensions and best approximation, in Parametric Optimization and Related Topics III, Gustrow 1991, P. Lang Verlag, Frankfurt am Main, pp. 107-120, 1993.

Dugundji, J., An extension of Tietze's theorem, Pacific J. Math., 1, pp. 353-367, 1951, https://msp.org/pjm/1951/1-3/pjm-v1-n3-p.pdf#page=35 DOI: https://doi.org/10.2140/pjm.1951.1.353

Dunford, N. and Schwartz, J. T., Linear Operators I, Interscience Publishers, New York, 1958.

Engelking, R., General Topology, PWN Warszawa, 1985.

Franić, I., An extension theorem for bounded linear 2-functionals, Math. Japon., 40, pp. 78-85, 1994.

Gähler, S., Linear 2-normierte Räume, Math. Nachr., 28, pp. 1-45, 1965, https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.19640280102 DOI: https://doi.org/10.1002/mana.19640280102

Harmand, P., Werner, D. and Werner, W., M-Ideals in Banach Spaces and Banach Algebras, Lect. Notes Mat., 1547, Springer-Verlag, Berlin, 1993, http://page.mi.fu-berlin.de/werner99/mideals.html DOI: https://doi.org/10.1007/BFb0084355

Hasumi, M., The extension property of complex Banach spaces, Tohoku Math. J., 10, pp. 135-142, 1958. DOI: https://doi.org/10.2748/tmj/1178244708

Holmes, R. B., Geometric Functional Analysis, Springer-Verlag, Berlin, 1975.

Xu Ji Hong, Norm-preserving extensions and best approximations, J. Math. Anal. Appl., 183, pp. 631-638, 1994, https://doi.org/10.1006/jmaa.1994.1170 DOI: https://doi.org/10.1006/jmaa.1994.1170

Kelley, J. L., Banach spaces with the extension property, Trans. Amer. Math. Soc., 72, pp. 323-326, 1952, https://doi.org/10.2307/1990758 DOI: https://doi.org/10.1090/S0002-9947-1952-0045940-5

Kirszbraun, M., Uber zusammenziehende Lipschitzsche Transformationen, Fund. Math., 22, pp. 77-108, 1934, http://matwbn.icm.edu.pl/ksiazki/fm/fm22/fm22112.pdf DOI: https://doi.org/10.4064/fm-22-1-77-108

Mabizela, S., The relationship between Lipschitz extensions, best approximation and continuous selections, Quaest. Math., 14, pp. 261-268, 1991, https://doi.org/10.1080/16073606.1991.9631644 DOI: https://doi.org/10.1080/16073606.1991.9631644

Mabizela, S., On bounded 2-linear functionals, Math. Japon., 35, pp. 51-55, 1990.

McShane, F. 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

Minty, G. J., On the extension of Lipschitz, Lipschitz-Holder continuous, and monotone functions, Bull. Amer. Math. Soc., 76, pp. 334-339, 1970, https://www.ams.org/journals/bull/1970-76-02/S0002-9904-1970-12466-1/S0002-9904-1970-12466-1.pdf

Mustăţa, C., Best approximation and unique extension of Lipschitz functions, J. Approx. Theory, 19, pp. 222-230, 1977, https://core.ac.uk/download/pdf/82250991.pdf

Mustăţa, C., Norm-preserving extensions of starshaped Lipschitz functions, Mathematica, 19, pp. 183-187, 1977. DOI: https://doi.org/10.1016/0021-9045(77)90053-3

Mustăţa, C., Extension of bounded Lipschitz functions and some related problems of best approximation, Babeş-Bolyai Univ., Seminar on Func. Anal. and Numerical Methods, Preprint no. 4, pp. 93-99, 1981.

Mustăţa, C., On the unicity of the extension of Lipschitz odd functions, Babeş-Bolyai University, Seminar on Optimization Theory, Preprint no. 8, pp. 75-80, 1987.

Mustăţa, C., M-ideals in metric spaces, Seminar on Mathematical Analysis, Babeş-Bolyai University, Preprint no. 7, pp. 65-74, 1988.

Mustăţa, C., Extension of Hölder functions and some related problems of best approximation, Seminar on Mathematical Analysis, Babeş-Bolyai University, Cluj-Napoca, Preprint no. 7, pp. 71-86, 1991.

Nachbin, L., A theorem of Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc., 68, pp. 28-46, 1950, https://doi.org/10.2307/1990536 DOI: https://doi.org/10.1090/S0002-9947-1950-0032932-3

Oja, E., On the uniqueness of the norm preserving extension of a linear continuous functional in the Hahn-Banach theorem, Izv. Akad. Nauk Est. SSSR Ser. Fiz. Mat., 33, pp. 424-438, 1984 (in Russian).

Oja, E., Extension of functionals and the structure of the space of continuous linear operators, Tartu University, Tartu, 1991 (in Russian).

Oja, E. and Poldvere, M., On subspaces of Banach spaces where every functional has a unique norm-preserving extension, Studia Math., 117, pp. 289-306, 1996.

Park, S. H., Quotient mappings, Helly extensions, Hahn-Banach extensions, Tietze extensions, Lipschitz extensions and best approximation, J. Korean Math. Soc., 29, pp. 239-250, 1992, https://www.koreascience.or.kr/article/JAKO199221048975373.pdf

Phelps, R. R., Uniqueness of Hahn-Banach extensions and best approximations, Trans. Amer. Marth Soc., 95, pp. 238-255, 1960, https://doi.org/10.2307/1993289 DOI: https://doi.org/10.1090/S0002-9947-1960-0113125-4

Rozema, E., Almost Chebyshev subspaces, lower semi-continuity and Hahn-Banach extensions, Proc. Amer. Math. Soc., 39, pp. 117-121, 1973, https://doi.org/10.1090/S0002-9939-1973-0312131-9 DOI: https://doi.org/10.1090/S0002-9939-1973-0312131-9

Singer, I., Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Ed. Academiei and Springer-Verlag, Bucureşti-Berlin, 1970. DOI: https://doi.org/10.1007/978-3-662-41583-2

Trenogin, V. A., Pisarevskij, E. M. and Soboleva, T. S., Problems and Exercises in Functional Analysis, Nauka, Moscow, 1984 (in Russian).

White, A. G., 2-Banach spaces, Math. Nachr., 42, pp. 43-60, 1969, https://onlinelibrary.wiley.com/doi/10.1002/mana.19690420104 DOI: https://doi.org/10.1002/mana.19690420104

Yosida, K. and Hewitt, E., Finitely additive measures, Trans. Amer. Math. Soc., 72, pp. 46-66, 1952, https://doi.org/10.1090/S0002-9947-1952-0045194-X DOI: https://doi.org/10.1090/S0002-9947-1952-0045194-X