On Best Approximation Problems In Normed Spaces With S-property

The problem of best approximation is the problem of finding, for a given point xX and a given set G in a normed linear space ( X, ), a point g G which should be nearest to x among all points of the set G.This thesis contains properties of best approximations in spaces with the S-property. We provide original results about Orlicz subspaces, and about subspaces with the S-property. As a major result we prove that: if G is a closed subspace of X and has the S-property. Then the following are equivalent:1.G is a Chebyshev subspace of X.2.L (m,G) is a Chebyshev subspace of L (m,X).3.L (m,G) is a Chebyshev subspace of L (m,X), 1pound