Will prove existence and uniqueness for approximation in the uniform norm in chapter

will prove existence and uniqueness for approximation in the uniform norm in chapter

To solve this problem, we want to minimize

since

Theorem 1. The best approximating polynomial is such that

if and only if

where

Thus, the integral is minimal if is the orthogonal projection of the function on the subspace . Suppose that form an orthogonal basis for . Then

Orthogonal polynomials can be obtained by applying the Gram-Schmidt Process to the basis for the inner product space .
3.1.1 The Gram-Schmidt Process