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...

For a closed subspace Y of a Banach space X, we define a separably determined property for Y in X. Let (P) be either proximinality or strong 1 1 2 -ball property and if (P) is separably determined for Y in X, then we prove that L1(μ, Y ) has the same property (P) in L1(μ,X). For an M -embedded space X, we give a class of elements in L1(μ,X ∗∗) having best approximations from L1(μ,X). We also pr...

Using two results of Garkavi, Medvedev and Khavinson [7], we give sufficient conditions for proximinality of sums of two ridge functions with bounded and continuous summands in the spaces of bounded and continuous multivariate functions respectively. In the first case, we give an example which shows that the corresponding sufficient condition cannot be made weaker for some subsets of Rn. In the...

As a counterpart to best approximation, the concept of best coapproximation was introduced in normed linear spaces by C. Franchetti and M. Furi in 1972. Subsequently, this study was taken up by many researchers. In this paper, we discuss some results on the existence and uniqueness of best approximation and best coapproximation when the underlying spaces are metric linear spaces. A new kind of ...

A metric space (X, d) is called an M-space if for every x and y in X and for every r 6 [0, A] we have B[x, r] Cl B[y, A — r] = {2} for some z € X, where A = d(x, y). It is the object of this paper to study M-spaces in terms of proximinality properties of certain sets. 0. Introduction. Let (X, d) be a metric space, and G be a closed subset of X. For x E X, let p(x,G) = inf{d(x, y) : y E G}. If t...

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...