ALTERNATION THEOREM FOR C(I,X) AND APPLICATION TO BEST LOCAL APPROXIMATION - AL-ZAMEL, A.; KHALIL, R. (Tamkang Journal of Mathematics, 1993)
Let X be a Banach space with the approximation property, and C(I,X) the space of continuous functions defined on I = [0,1) with values in X. Let u_i C(I,X), i=1,2,…, n and M=span{u_1, …, u_n}. The object of this paper is to prove that if {u_1, …, u_n} satisfies certain conditions, then for f C(I,X) and g M we have ||f-g|| = inf||f-h|| : hM} if and only if f-g has at least n-zeros. An application to best local approximation in C(I,X) is given.