FIVE MAJOR RESULTS IN ANALYSIS AND TOPOLOGY Aaron ...

CHAPTER 3
The Stone-Weierstrass Theorem
One of the most important structures studied in mathematical analysis is the set C(X,Y )
of all continuous functions from a compact space X (such as [0, 1]) into a metric space Y
(such as R or C). An understanding of this set is crucial for the development of modern
and classical calculus, differential equation theory, differential geometry, and several other
branches of mathematics. For many applications in classical analysis, X is taken to be a
closed interval of the real line, and Y is taken to be either C or, as a special case, R. One
particular subset of C([0, 1], C) - the class of all complex polynomials on [0, 1] - is particularly
easy to study. Karl Weierstrass showed that, in the metric space (C([0, 1], C),ρ∞), for any
function f and any ǫ > 0, one can find a polynomial p such that ρ∞(f,p) < ǫ. This result is
known as the Weierstrass approximation theorem. There are two widely known constructive
proofs of this theorem. The version presented here uses a set of polynomials known as
the Landau kernels, and follows that given in [12]. The other proof uses the Bernstein
polynomials, and can be found in [13].
Theorem (Weierstrass). Let f be a continuous complex-valued function on [a,b]. Then
there exists a sequence of polynomials Pn such that Pn(x) → f(x) uniformly on [a,b]. If f
is real, then the Pn can be taken to be real.
Proof. Without loss of generality, assume that [a,b] = [0, 1], and that f(0) = f(1) = 0.
Furthermore, define f(x) to be zero for every x /∈ [0, 1]. Then f is uniformly continuous on
the entire real line. Let Qn(x) = cn(1 − x 2 ) n for each n ∈ N, where cn is chosen to ensure
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