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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