On the converses of the reduction principle in inner product spaces

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 \(X\).

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

Inner product spaces; the Reduction Principle; best approximation.

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

http://www.cs.ubbcluj.ro/~studia-m/2006-3/mustata.pdf

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