Simultaneously proximinal subspaces

• Autorzy: Rao T. S. S. R. K.

Identyfikatory

DOI

10.1515/jaa-2016-0012

• Języki publikacji: EN

Abstrakty

EN

In this paper we study closed subspaces of Banach spaces that admit relative Chebyshev centers for all bounded subsets of the space. We exhibit new classes of spaces which have this property and study stability results similar to the ones studied in the literature in the context of proximinal subspaces and Chebyshev centers. For the space C(K) of continuous functions on a compact set K, we show that a closed subspace of finite codimension has relative Chebyshev centers for all bounded sets in C(K) if and only if it is a strongly proximinal subspace.

Słowa kluczowe

EN

simultaneously proximinal subspaces; Chebyshev radius; Chebyshev centers

PL

podprzestrzeń; promień Czebyszewa; środek Czebyszewa

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2016
• Tom: Vol. 22, nr 2
• Strony: 115--120
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Rao T. S. S. R. K.

• Theoretical Statistics and Mathematics Unit, Indian Statistical Institute,R. V. College P.O., Bangalore 560059, India

Bibliografia

• [1] P. Bandyopadhyay and T. S. S. R. K. Rao, Central subspaces of Banach spaces, J. Approx. Theory 103 (2000), 206-222.
• [2] J. R. Catder, W. P. Coleman and R. L Harris, Centers of infinite bounded sets in a normed space, Canad. J. Math. 25 (1973), 986-999.
• [3] E. W. Cheney and D. E. Wulbert, The existence and unicity of best approximations, Math. Scand. 24 (1969), 113-140.
• [4] S. Dutta and D. Narayana, Strongly proximinal subspaces of finite codimension in C(K), Colloq. Math. 109 (2007), 119-128.
• [5] V. Indumathi, Proximinal subspaces of finite codimension in direct sum spaces, Proc. Indian Acad. Sci. Math. Sci. Ill (2001), 229-239.
• [6] C. R. Jayanarayanan and T. Paul, Strong proximinality and intersection properties of balls in Banach spaces, J. Math. Anal. Appl. 426 (2015), 1217-1231.
• [7] H. E. Lacey, The Isometric Theory of Classical Banach Spaces, Grundlehren Math. Wiss. 208, Springer, Berlin, 1974.
• [8] W. A. Light and E. W. Cheney, Approximation Theory in Tensor Product Spaces, Lecture Notes in Mathematics 1169, Springer, Berlin, 1985.
• [9] P.-K. Lin, A remark on the sum of proximinal subspaces, J. Approx. Theory 58(1989), 55-57.
• [10] J. Mendoza and T. Pakhrou, Best simultaneous approximation in L1(μ,X) J. Approx. Theory 145 (2007), 212-220.
• [11] T. S. S. R. K. Rao, Best constrained approximation in Banach spaces, Numer. Fu net Anal. Optim. 36 (2015), 248-255.
• [12] M. Rawashdeh, Sh. Al-Sharif and W. B. Domi, On the sum of best simultaneously proximinal subspaces, Hacet J. Math. Stat. 43 (2014), 595-602.
• [13] F. B. Saidi, D. Hussein and R. Khalil, Best simultaneous approximation in Lp(l, E), J. Approx. Theory 116 (2002), 369-379.
• [14] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, GrundlehrenMath.Wiss.171, Springer, Berlin, 1970.
• [15] P. W. Smith and J. D. Ward, Restricted centers in C(Ω), Proc. Amer. Math. Soc. 48 (1975), 165-172.
• [16] L. Vesely, A Banach space in which all compact sets, but not all bounded sets, admit Chebyshev centers, Arch. Math. (Basel) 79 (2002), 499-506.
• [17] L. Vesely, Chebyshev centers in hyperplanes of c0, Czechoslovak Math. J. 52(127) (2002), 721-729.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.