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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
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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

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References

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