Abstract: Let X be a Banach space and Y a closed convex subset of X containing the original.The following is the main result of this paper:Lp(μ,Y) is a Chebyshev subset of Lp(μ,X) (1<p<∞) if and only if L1(μ,Y) is a Chebyshev subset of L1(μ,X).In addition,this paper gives an example to show that the conclusion “that g is a best approximation to f∈L∞(μ,X) from L∞(μ,Y) implies g(s)∈ PY(f(s))　for almost all s∈Ω” is not true.

Key words: best approximation, Chebyshev subset, closed convex set