Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-05-16T13:59:29.732Z Has data issue: false hasContentIssue false
  • English
  • Français

Norms in product spaces which preserve approximation properties

Published online by Cambridge University Press:  14 November 2011

Carlos Benítez  and
Manuel Fernández
Carlos Benítez
Affiliation:
Departamento de Matemáticas, Universidad de Extremadura, 06071 Badajoz, Spain
Manuel Fernández
Affiliation:
Departamento de Matemáticas, Universidad de Extremadura, 06071 Badajoz, Spain
Get access Rights & Permissions [Opens in a new window]

Synopsis

Let E1 and E2 be real normed linear spaces such that the dimension of any of them is at least 2. We prove that the norms in E1 × E2 which verify a simple property of monotonicity with regard to the initial norms in E1 and E2 are the only norms in E1 × E2 which preserve best linear approximations, in the sense that ifyk ∊ Lk is best approximation to xk from the linear subspace Lk, (k = 1,2), then (y1, y2) is best approximation to (x1, x2) from L1 × L2.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 105 , Issue 1 , 1987 , pp. 199 - 203
Copyright
Copyright © Royal Society of Edinburgh 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Benitez, C.. Normas en espacios producto que conservan propiedades de aproximación. Rev. Mat. Hisp.-Amer. 4 34 (1974), 163–175.Google Scholar
2Birkhoff, G.. Orthogonality in linear metric spaces. Duke Math. J. 1 (1935), 169–172.CrossRefGoogle Scholar
3Cartan, H.. Calcul Différentiel (Paris: Hermann, 1967).Google Scholar
4Hammer, P. C. and Soboczyk, A.. Planar line families I. Proc. Amer. Math. Soc. 4 (1953), 226–233.CrossRefGoogle Scholar
5James, R. C.. Orthogonality and linear functionals in normed linear spaces. Trans. Amer. Math. Soc. 6 (1947), 265–292.CrossRefGoogle Scholar
6Singer, I.. Best approximation in normed linear spaces by elements of linear subspaces (Berlin: Springer-Verlag, 1970).CrossRefGoogle Scholar