On continuous selections for metric projections in spaces of continuous functions

AbstractX is a compact Hausdorff space and C(X) the Banach space of real-valued continuous functions on X. Amongst other results it is shown that, if M is a closed linear subspace of C(X) such that no nonzero member of M is zero on a nonempty open subset of X and for each f in C(X) the metric projection Pm(f) of f onto M is nonempty and finite-dimensional, then if there is a continuous selection for PM it is unique. An example is given of a five-dimensional subspace M of C([−1, 1]) which is non-Chebyshev and for whose metric projection PM there is a unique continuous selection. This example shows that a result claimed by other authors in a previous paper on this subject is false