The Stone-Weierstrass Theorem

The Weierstrass Approximation Theorem shows that the continuous realvalued fuctions on a compact interval can be uniformly approximated by polynomials. In other words, the polynomials are uniformly dense in C([a, b],R) with respect to the sup-norm. The original proof was given in [1] in 1885. There are now several different proofs that use vastly different approaches. One well-known proof was given by the Russian Sergei Bernstein in 1911. His proof uses only elementary methods and gives an explicit algorithm for approximating a function by the use of a class of polynomials now bearing his name. It will be seen that the Weierstrass Approximation Theorem is in fact a special case of the more general Stone-Weierstrass Theorem, proved by Stone in 1937, who realized that very few of the properties of the polynomials were essential to the theorem. Although this proof is not constructive and relies on more machinery than that of Bernstein, it is much more efficient and has the added power of generality. First, Bernstien’s proof of the Weierstrass Approximation Theorem, which is taken from [4], is examined. The Bernstein polynomials, which play a central role in Bernstein’s proof, are introduced below.