578 Chapter 9 • Sequences and Series of Functions
Corollary 9.4.20 Given any continuous f : [a, b] ---+ JR, there is a series
00
L Pn(x) of polynomials converging uniformly to f on [a,b].
k=l
Proof. See Exercise 8.1.6. •Weierstrass' approximation theorem may be regarded as the fundamental
theorem of "approximation theory," an area of mathematical research that
came into prominence with the rise of computing technology. This area of
study seeks to calculate the values of complicated functions by using simpler
(more easily calculated) functions called "approximations." Polynomials are
often used as approximations because of their relative simplicity. Although
mathematicians of our time have developed a greater variety of approximation
methods, polynomials remain a valuable tool in their arsenal.
A more popular approach to proving the Weierstrass approximation theo-
rem uses "Bernstein polynomials." Recommended readings for this approach
are Douglass [32] pp. 16 8- 172, Gaskill and Narayanaswami [47] pp. 428-435,
and Pugh [110] pp. 217 - 222. Another approach using "Dirac sequences" may
be found in Douglass [32] pp. 281 - 285 , Hairer and Wanner [61] pp. 265 - 269,
and Stoll [128] pp. 346-352. For other approaches see the comprehensive survey
paper by Allan Pinkus [107].
9.5 *A Glimpse Beyond the Horizon
It is time to confess that a thorough understanding of the Elements of Real
Analysis is not an end in itself. It is rather a beginning. It opens the door to
further study of a wide panorama of areas of modern analysis, such as multi-
variable real analysis, general integration theories, complex analysis, functional
analysis, Fourier analysis, approximation theory, and various areas of applied
mathematics.
We close this course with a few tantalizing ideas that may whet your ap-
petite for further study in real and functional analysis. To do justice to these
ideas would require several chapters, so we shall have to settle for a quick
glimpse beyond the horizon.
Definition 9.5.1 A normed vector space is a vector space V together with
a "norm" II· II; that is , a function from V to JR such that Vu, v E V, and \Ir E JR,
(a) llull ~ 0, and llull = 0 if and only if u = O;
(b) llu +vii :::; llull + llvll (triangle inequality);
( c) llrull = lrl llull-
In Theorem 9.2.2 we saw that the "s up norm" defined in 9.2.l has these
properties on the space of bounded functions on [a, b], as well as on subspaces