Metric projections in spaces of integrable functions (Q1890587)

Interest in properties of set valued metric projections arose from attempts to find single valued continuous selections. This work concentrates on the properties of metric projections onto finite dimensional subspaces of spaces of integrable functions. A mapping called ``the derived metric projection'' is constructed. Special cases of the theory developed here are used to recover the best known results. For example, the lower semicontinuous metric projections are completely characterized in this setting. Examples are presented showing that the situations for real and for complex valued functions are fundamentally different. This paper, as well as continuing the list of fundamental contributions by Professor Brown, is also closely related to the 1991 work of \textit{W. Li} [Trans. Am. Math. Soc. 322, No. 2, 583-591 (1990; Zbl 0728.41020)].