Minima and best approximations in constructive analysis

Working in Bishop’s constructive mathematics, we first show that minima can be defined as best approximations, in such a way as to preserve the compactness of the underlying metric space when the function is uniformly continuous. Results about finding minima can therefore be carried over to the setting of finding best approximations. In particular, the implication from having at most one best approximation to having uniformly at most one best approximation is equivalent to Brouwer’s fan theorem for decidable bars. We then show that for the particular case of finite-dimensional subspaces of normed spaces, these two notions do coincide. This gives us a better understanding of Bridges’ proof that finite-dimensional subspaces with at most one best approximation do in fact have one. As a complement we briefly review how the case of best approximations to a convex subset of a uniformly convex normed space fits into the unique existence paradigm.