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Let X be a Banach space, (I, å , μ) a finite measure space and L¹(μ,X) the Banach space of all X-valued μ-integrable functions on the unit interval I equipped with the usual 1-norm. In this paper we prove that for a closed subspace G of X, L¹(μ,G) is simultaneously Chebyshev in L¹(μ,X) if and only if G is simultaneously Chebyshev in X. Further results are obtained in the space of bounded linear operators L(l¹,X) and in the space of continuous functions C¹(I, l^p) with respect to the L¹ norm.

Mathematics Subject Classification (2010): 41A65, 41A50.

Keywords: Best approximation, simultaneous approximation, spaces of vector functions.