In this chapter we begin the study of best approximations. In this case we study the best (min) polynomial approximation in the uniform (max) norm. The existence a best approximating polynomial is first presented. The more subtle issue of uniqueness is then discussed. To show uniqueness the celebrated de la Vallee Poussin, and Chebyshev equi-oscillation theorems are presented. A first error estimate is then presented. The problems of interpolation, discussed in the previous chapter, and best approximation are then related via the Lebesgue constant. Chebyshev polynomials are then introduced, and their most relevant properties presented. Interpolation at Chebyshev nodes, and the mitigation of the Runge phenomenon are then discussed. Finally; Bernstein polynomials; moduli of continuity and smoothness; are detailed in order to study Weierstrass approximation theorem.

We prove some key results about spaces of continuous functions. First we show that continuous functions on an interval can be uniformly approximated by polynomials (the Weierstrass Approximation Theorem), which has interesting applications to Fourier series. Then we prove the Stone-Weierestrass Theorem, which generalises this to continuous functions on compact metric spaces and other collections of approximating functions. We end with a proof of the Arzelà-Ascoli Theorem.