## Abstract

Let \(H\) be an inner product space, \(X\) a complete subspace of \(H\), and \(Y\) a closed subspace of \(X\). The main result of this Note is the following converse of the Reduction Principle: if \(x_{0}\in X,\ h\in H\backslash X\) and \(y_{0}\in Y\) is the element of best approximation of both \(x_{0}\) and \(h\), \((x_{0}-h,x_{0}-y_{0})=0\) and \(codim_{X}Y=1\), then \(x_{0}\) is the element of best approximation of \(h\) in \(X\).

## Authors

**Costică Mustăţa**

“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

## Keywords

Inner product spaces; the Reduction Principle; best approximation.

## Paper coordinates

C. Mustăţa, *On the converses of the reduction principle in inner product spaces,* Studia Univ. Babeş-Bolyai, Mathematica, **51** (2006) no. 3, 97-104.

## About this paper

##### Journal

Mathematica

##### Publisher Name

Studia Universitatis Babes-Bolyai, Mathematica

##### DOI

##### Print ISSN

1843-3855

##### Online ISSN

2065-9490

google scholar link

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[2] Deutsch, F., Best Approximation in Inner-Product Space, Springer-Verlag, New YorkBerlin-Heidelberg, 2001.

[3] Laurent, P.J., Approximation et Optimisation, Herman, Paris, 1972.