Existence of best approximation elements in $C(Q,X)$

Generalizing the result of A. L. Garkavi (the case $X=\mathbb R$) and his own previous result concerning $X=\mathbb C$), the author characterizes the existence subspaces of finite codimension in the space $C(Q,X)$ of continuous functions on a bicompact space $Q$ with values in a Banach space $X$, under some assumptions concerning $X$. Under the same assumptions, it is proved that in the space of uniform limits of simple functions, each subspace of the form
$$ \biggl\{g\in B:\int_Q\bigl\langle g(t),d\mu_i\bigr\rangle=0,\ i=1,\dots,n\biggr\}, $$
where $\mu_i\in C(Q,X)^*$ are vector measures of regular bounded variation, is an existence subspace (the integral is understood in the sense of Gavurin).