The conditional Chebyshev center of a compact set of continuous functions

We establish characteristic properties of a subspace $L$ of finite codimension of the space $C(T)$ that has a Chebyshev center and a Chebyshev net for every compact set from $C(T)$. We show that these properties are the same as the conditions for the existence in $L$ of an element of best approximation for every element from $C(T)$.