Best approximation in vector valued function spaces

Let T be the unit circle, and m be the normalized Lebesgue measure on T. If H is a separable Hilbert space, we let L^∞T,H) be the space of essentially bounded functions on T with values in H. Continuous functions with values in H are denoted by C(T,H), and H^∞(T,H) is the space of bounded holomorphic functions in the unit disk with values in H. The object of this paper is to prove that (H^∞+C)(T,H) is proximinal in L^∞(T,H). This generalizes the scalar valued case done by Axler, S. et al. We also prove that (H^∞+C)(T,l^∞) |H^∞(T,l^∞) is an M-ideal of L^∞(T,l^∞) | H^∞ (T, l^∞), and V(T,l^∞) is an M-ideal of L^∞(T, l^∞)whenever V is an M-ideal of L^∞, where V(T,l∞) {g ϵ L^∞(T,l^∞): <g(t), δ_n > ϵ V for all n}.