Abstract
In this chapter, we consider classical ideas and results that formed the basis of the theory of approximation by finite-dimensional subspaces. In spaces C(Q), we give several results that either characterize or give sufficient conditions for the existence of Chebyshev subspaces in C(Q). Among such conditions, we mention de la Vallée Poussin’s estimates (see Sect. 2.1), the Haar characterization property (see Sect. 2.3), and Mairhuber’s theorem (see Sect. 2.5), which characterizes the metrizable compact sets Q such that the space C(Q) contains nontrivial finite-dimensional Chebyshev subspaces.
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