Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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BEST APPROXIMATIONS IN $L_{p}$(S,X)

  • Published : 1999.08.01
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Abstract

Let G be a closed subspace of a Banach space X and let (S,$\Omega$,$\mu$) be a $\sigma$-finite measure space. It was known that $L_1$(S,G) is proximinal in $L_1$(S,X) if and only if $L_p$(S,G) is proximinal in $L_p$(S,X) for 1$\infty$. In this article we show that this result remains true when "proximinal" is replaced by "Chebyshev". In addition, it is shown that if G is a proximinal subspace of X such that either G or the kernel of the metric projection $P_G$ is separable then, for 0 < p $\leq$ $\infty$. $L_p$(S,G) is proximinal in $L_p$(S,X)

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References

  1. Math. Proc. Camb. Phil. Soc. v.104 Best approximation in L(X,Y) W. Deeb;R. Khalil
  2. Ann. Mat. Pure Appl. v.101 Multifunctions on abstract measurable spaces and applications to stochastic decision theory C. J. Himmelberg;F. S. Van Vleck
  3. Soochow J. Math. v.18 no.4 Best approximation in tensor product spaces D. Hussein:R. Khalil
  4. Math. Proc. Camb. Phil. Soc. v.94 Best approximation in L$^p$(I, X) R. Khalil
  5. J. Approx. Theory v.59 Best approximation in L$^p$(μ, X)Ⅱ R. Khalil;W. Deeb
  6. Proc. Amer. Math. Soc. v.123 Best approximation in L$^l$(I, X) R. Khalil:F. Saidi
  7. Bull. Korean Math. Soc. v.94 Proximity maps for certain spaces M. B. Lee;S. H. Park
  8. Rocky Mountain J. Math. v.19 Proximinality in L$_p$(S, Y) W. A. Light
  9. Lect. Notes in Math. v.1169 Approximation theory in tensor product spaces W. A. Light;E. W. Cheney
  10. J. Approx. Theory v.83 On the smoothness of the metric projection and its application to proximinality in L$^p$(S, X) F. B. Saidi
  11. J. Approx. Theory v.78 Point-wise best approximation in the space of strongly measurable functions with applications to best approximation in L$^p$(μ, X) You Zhao-Yong;Guo Tie-Xin