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Existence of best approximation elements in $C(Q,X)$
L. P. Vlasov Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences
Abstract:
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).
Received: 04.04.1994
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