## Remarks on remotal sets in vector valued function spaces.(English)Zbl 1207.46018

A subset $$E$$ of a Banach space $$X$$ is called remotal if for any $$x\in X$$ there is a remotal point in $$E$$. The present paper is concerned with remotal sets in the space $$L^p([0,1],X)$$ of $$p$$-Bochner integrable functions (classes). In particular, it is proved that, if $$E$$ is a finite set in a Banach space $$X$$, then $$L^1([0,1],E)$$ is remotal in $$L^1([0,1],X)$$; if $$E$$ is a closed bounded set in $$X$$ such that the span of $$E$$ is a finite-dimensional subspace of $$X$$, then $$L^1([0,1],E)$$ is remotal in $$L^1([0,1],X)$$; also, if $$E$$ is a closed bounded subset of a Banach space $$X$$, then $$L^1([0,1],E)$$ is remotal in $$L^1([0,1],X)$$ if and only if $$L^p([0,1],E)$$ is remotal in $$L^p([0,1],X)$$, $$1 < p < \infty$$.

### MSC:

 46B20 Geometry and structure of normed linear spaces 41A50 Best approximation, Chebyshev systems 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

### Keywords:

remotal sets; approximation theory in Banach spaces
Full Text: