Proximinality in generalized direct sums

We consider proximinality and transitivity of proximinality for subspaces of finite codimen-sion in generalized direct sums of Banach spaces. We give several examples of Banach spaces where proximinality is transitive among subspaces of finite codimension. 1. Introduction. Let X be a Banach space and let Y be a closed subspace of X. We recall that Y is said to be a proximinal subspace of X if for any x ∈ X there exists a y ∈ Y such that d(x, Y) = =x − y. In the first part of the paper, we study proximinal subspaces of finite codimension in generalized direct sums of Banach spaces (a concept due to Veselý [9], see below for the definition). Our motivation comes from some recent work of Indumathi [4] where she considered these questions for c 0-direct sums of a family of Banach spaces and proved the following.