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Sheltered points in normed spaces

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Summary

If X is a Hilbert space and V is a subspace, given x ε X/V the best approximation IIv(x) to x from V can be constructed in the following way: we consider a ball Br centered at x and intersecting V, then we take the (Chebyshev) center of Br ∩ V. In Banach spaces the centers in V of Br ∩ V (which depend on r) are related to a different map of « approximation ». Here we introduce the notions of « sheltered point » and « shelter » of a bounded set, and we obtain in Banach spaces information about IIv(x) starting from the shelter of Br ∩ V. The notion of sheltered point turns out to be of some interest by itself, since translates problems of simultaneous worst approximation.

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Entrata in Redazione il 1° giugno 1977.

Work performed under the auspices of the CNR (Consiglio Nazionale delle Ricerche).

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Papini, P.L. Sheltered points in normed spaces. Annali di Matematica 117, 233–242 (1978). https://doi.org/10.1007/BF02417893

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Keywords

  • Hilbert Space
  • Banach Space
  • Normed Space
  • Space Information
  • Sheltered Point