Coproximinality in Spaces of Bochner Integrable Functions

In this short note we use a new way of applying von Neumann's selection theorem for obtaining best coapproximation in spaces of measurable functions. For a coproximinal closed subspace

of a Banach space

X X

, we show that if

Y Y

is constrained in a weakly compactly generated dual space, then the space

L^1(\mu,Y) L 1 ( μ , Y )

of

Y Y

-valued Bochner integrable functions is coproximinal in

L^1(\mu,X) L 1 ( μ , X )

. This extends a result of M. R. Haddadi, N. Hejazjpoor and H. Mazaheri [Some results about best coapproximation in

L^P(S,X) L P ( S , X )

, Anal. Theory Appl. 26 (2010) 69–75], proved when

Y Y

is reflexive.