ALTERNATION THEOREM FOR $C(I,X)$ AND APPLICATION TO BEST LOCAL APPROXIMATION

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 $u_i \in C(I,X)$, $i=1,2,\cdots, n$ and $M=span\{u_1, \cdots, u_n\}$. The object of this paper is to prove that if $\{u_1, \cdots, u_n\}$ satisfies certain conditions, then for $f \in C(I,X)$ and $g \in M$ we have $||f-g|| = \inf\{||f-h|| : h\in M\}$ if and only if $f-g$ has at least $n$-zeros. An application to best local approximation in $C(I,X)$ is given.