We characterize finite-dimensional normed linear spaces as strongly proximinal subspaces in all their superspaces. A connection between upper Hausdorff semi-continuity of metric projection and finite dimensionality of subspace is given.

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 f...

We derive transitivity of various degrees of proximinality in Banach spaces. When the transitivity does not carry forward to the bigger space we investigate these properties under some additional assumptions of the intermediate space. For instance, we show that if Z ⊆ Y ⊆ X where Z is a finite co-dimensional subspace of X which is strongly proximinal in Y and Y is an M-ideal in X then Z is stro...

In a normed linear space X, consider a nonempty closed set K which has the property that for some r > 0 there exists a set of points xo € X\K, d(xoK) > r, which have closest points p(xo) € K and where the set of points xo — r((xo — p(xo))/\\xo — p(zo)||) is dense in X\K. If the norm has sufficiently strong differentiability properties, then the distance function d generated by K has similar dif...

Lifting properties for Banach spaces are studied. An alternate version of the lifting property due to Lindenstrass and Tzafriri leads a new geometric called quotient pairs (X, J), with J proximinal in X. Several necessary sufficient conditions hold given.