NONLINEAR PROJECTIONS AND GENERALIZED CONDITIONAL EXPECTATIONS IN BANACH SPACES

Abstract

Let $E$ be a smooth, strictly convex and reflexive Banach space, let $C_*$ be a closed linear subspace of the dual space $E^*$ of $E$ and let $\Pi_{C_*}$ be the generalized projection of $E^*$ onto $C_*$. In this paper, we study the mapping $R$ defined by $R = J^{-1} \Pi_{C_*}J$, where $J$ is the normalized duality mapping from $E$ into $E^*$. We obtain some results which are related to conditional expectations and martingales in the probability theory.