Characterizations of the metric and generalized metric projections on subspaces of Banach spaces

Abstract

The variational principles of the metric projection onto closed and convex sets are commonly used as the basis for characterizations of metric projections onto subspaces of Hilbert spaces. However, it is important to realize that the variational principles do not hold as necessary and sufficient conditions in general Banach spaces to characterize the metric projection and its related extensions. This technical handicap prevents the study of metric projection onto subspaces of general Banach spaces from being fully explored. Motivated by this existing lacuna in the literature, we investigate the metric projection and the generalized metric projection onto subspaces of general Banach spaces in this study. Focusing on concrete Banach spaces, we show that even though there are no general variational principles in such spaces for the projection onto subspaces, specific elements satisfy a variational characterization. We provide concrete examples to illustrate various notions of projections in general Banach spaces.