The problem of best approximation is the problem of finding, for a given point xÎX and a given set G in a normed linear space ( X, ), a point g Î 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), 1£ p <¥.

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