Best Approximation in Normed Linear Spaces by Elements of Arbitrary Linear Subspaces

Abstract

In the present paragraph we shall give characterizations of elements of best approximation and some consequences of these characterizations in arbitrary (complex or real) normed linear spaces, and we shall apply them to various concrete spaces. Since we have *)

$$ {L_G}\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {x\quad for\quad x \in G} \\ {\quad for\quad x \in \overline {G\backslash G,} } \end{array}} \right. $$

(1.1)

for any linear subspace G of a normed linear space E, it will be sufficient to characterize the elements of best approximation of the elements $x \in E\backslash \bar G$. In order to exclude the trivial case when such elements x do not exist, throughout the sequel by “linear subspace” G⊂E we shall understand “proper linear subspace G which is not dense in E”, that is, we shall assume, without special mention, that $\bar G \ne E$.