Best approximation in vector valued function spaces (1985)
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∞): and lt;g(t), δn and gt; ϵ V for all n}.