Continuous Selections in C(X)

Suppose C(X)is a Banack space of real-valued continuous functions on X.For a finite-dimensional subapace G C(X),define d(f; G)=inf{||f-g||:g∈G}; P_G(f)={g∈G: ||f-g||=d(f;G)}. If there exists a continuous mapping S from C(X)to G such that S(f)∈P_G(f)for every f∈C(X),then we say P_G has a continuous selection. In this paper,we give several characterizations of P_G with a continuous sele-ction.