A note on proximinality in function spaces F(S,X)

We discuss initially how to construct a generalisation of the Lebesgue-Bochner function spaces by using a Banach lattice of real-valued functions and an appropriate lifting of the norm from this space to the more general setting. This lifting is shown to be well-defined, but the primary purpose is to discuss the notion of proximinality in this new space. In this paper, the results are derived by establishing the property of uniform convexity under suitable hypotheses. Other techniques may be found in the literature, and citations are given in the paper.