Geodesic
Existence of best approximation elements in $C(Q,X)$
Matematičeskie zametki, Tome 58 (1995) no. 2, pp. 163-175.

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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). 
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L. P. Vlasov. Existence of best approximation elements in $C(Q,X)$. Matematičeskie zametki, Tome 58 (1995) no. 2, pp. 163-175. http://geodesic.mathdoc.fr/item/MZM_1995_58_2_a0/

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