FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

3. THE STONE-WEIERSTRASS THEOREM 21
f = λg and f = u + iv, then u(x) > 0, so that AR vanishes at no point of K. By the real
version of the Stone-Weierstrass theorem, the uniform closure of AR in A is C(K, R), and is
contained therefore in B, the uniform closure of A. The result follows from noting that B is
a complex algebra, and that if f ∈ C(K, C) then f = u + iv is in B because u,v are in the
uniform closure of AR ⊂ B.
There is an even further generalization of the Stone-Weierstrass theorem, which was
proved by Erret Bishop in 1961. The result, Bishop’s Theorem, can be found in [1] and [11].
As discussed before, the Stone-Weierstrass theorem and its special cases appear through-
out analysis, but primarily in results that involve maps into scalar fields. We will conclude
by providing a few interesting results which rely on this theorem.
The following extension of the Weierstrass approximation theorem which, although it is a
trivial corollary of the Stone-Weierstrass theorem, can be used to extend many of the results
of one-dimensional analysis into two dimensions.
Theorem (Extension of Weierstrass Approximation Theorem 1 ). If K ⊂ R 2 is compact,
and if f ∈ C(K, C), then there is a polynomial g(x1,x2) such that |f(x1,x2) − g(x1,x2)| < ǫ
for every (x1,x2) ∈ K.
The following result, from functional analysis, in incredibly useful. The n th moment of a
real continuous function f defined on [0, 1] is the value of the integral
1
0
f(x)x n dx.
Theorem (Moments of Real Functions 2 ). Let f,g be real continuous functions defined
on [0, 1]. Then f and g are identical if, and only if, the n th moments of f and g agree for
every n ∈ N.
This last result implies the following theorem.
Theorem (Number of Continuous Real Functions). The cardinality of the set of contin-
uous real functions defined on [0, 1] is equivalent to that of R.
1 Taken from [13].
2 Taken from [13].