In this particular section, real functions of are considered. A theorem due to Weierstrass (see for instance ) states that the set of all polynomials is a dense subspace of all the continuous functions on , with the norm . This maximum norm is defined as
This means that, for any continuous function of , there exists a sequence of polynomials that converges uniformly towards :
Given a continuous function , the best polynomial approximation of degree , is the polynomial that minimizes the norm of the difference between and itself:
Chebyshev alternate theorem states that for any continuous function , is unique (theorem 9.1 of  and theorem 23 of ). There exist points such that the error is exactly attained at those points in an alternate manner:
Living Rev. Relativity 12, (2009), 1
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