STONE-WEIERSTRASS THEOREM. BRUNO BROSOWSKI AND FRANK DEUTSCH1. Abstract. An elementary proof of the Stone-Weierstrass theorem is given.
proceedings of the american mathematical
society
Volume 81, Number 1, January 1981
AN ELEMENTARY PROOF OF THE STONE-WEIERSTRASS THEOREM BRUNO BROSOWSKI AND FRANK DEUTSCH1 Abstract.
An elementary proof of the Stone-Weierstrass theorem is given.
In this note we give an elementary proof of the Stone-Weierstrass theorem. The proof depends only on the definitions of compactness ("each open cover has a finite subcover") and continuity ("the inVerse images of open sets are open"), two simple consequences of these definitions (i.e. "a closed subset of a compact space is compact," and "a positive continuous function on a compact set has a positive infimum"), and the elementary BernoulU inequality:
(I + h)n > I + nh
(n = 1, 2, . . . )
if h > -1. In the beautiful and elementary proof of the classical Weierstrass theorem given by Kuhn [1], it is observed that it suffices to be able to approximate, by polynomials, the step function which is 1 on the interval [0, |) and 0 on the interval [\, 1] uniformly outside of each neighborhood of |. The main step in our proof (Lemma 2) is the general analogue of this. It shows that it suffices to be able to approximate, by elements of the subalgebra, a given "generalized step function" (i.e. a function which is 0 on a closed set and 1 off the set) uniformly on the closed set and off a
neighborhood of this set. It should be remarked that when our proof is specialized to the classical case of polynomials in C[a, b], it is even "simpler" than Kuhn's proof in the sense that no "change of variables" argument is necessary, nor is it necessary to appeal to the fact that continuous functions on [a, b] are uniformly continuous. (Kuhn's proof also used the BernoulU inequality.) In particular, it is perhaps worth emphasizing that, in contrast to many proofs of the Stone-Weierstrass theorem, we do not appeal to any of the foUowing facts: (a) the classical Weierstrass theorem (nor even the special case of uniformly approximating/(/) = |r| on [— 1, 1] by polynomials); (b) that the closure of a subalgebra is a subalgebra; (c) that the closure of a subalgebra is a sublattice. Let F be a compact topological space and C(T) the set of all real-valued continuous functions on T. A neighborhood of a point in F is an open set which Received by the editors July 10, 1979 and, in revised form, April 24, 1980. 1980 Mathematics Subject Classification. Primary 46J30, 46E15; Secondary 41A30. Key words and phrases. Stone-Weierstrass theorem, Weierstrass theorem, approximation by subalgebras, polynomial approximation. 'On leave from the Pennsylvania State University, University Park, Pennsylvania 16802. © 1981 American
Mathematical
0002-99 39/81/0000-0019/$02.00
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Society
90
BRUNO BROSOWSKI AND FRANK DEUTSCH
contains the point. Let 91be a subset of C(T) with the properties: (i) x, y in 91,a, ß in R implies ax + ßy G 81;
(ü) x,y in 91impUes jc • v e 91; (in) 1 G 9Í; (iv) if r, and t2 are distinct points in T, then there exists x G 91 such that *('i)
*
*('2)-
In other words, 91 is a "subalgebra of C(F) which contains constants and separates points." The Stone-Weierstrass theorem may be stated as follows. If 91 is a subalgebra of C(T) which contains constants and separates points, then the elements of C(T) can be uniformly approximated by the elements of 91. That is, given / G C(T) and e > 0 there exists g G 91 such that sup,eT|/(/) — g(t)\ < e. It is convenient to divide the proof into three steps. The essential step is Lemma 1. For brevity, the norm notation ||x|| = sup{|x(f)| |i e T) wül sometimes be used.
Lemma 1. Let t0 G T and let U be a neighborhood of t0. Then there is a neighborhood V = V(t0) of Iq, V g U, with the following property. For each e > 0, there exists x G 91 such that
(1) 0 < x(t) < l,t G T; (2)x(t) I - e,t GT\U. Proof. For each t G T \ U, the point separating property (iv) implies that there is a function g, G 91 with g,(t) ¥= g,('o)- Then the function h, = g, — g,(to) ■ 1 is in 91
and h,(t) ¥• h,(t0)= 0. The functionpt = (l/\\hf\\)hf is in 91,p,(íq) = 0, p,(t) > 0, and 0 < pt < 1. Let U(t) = {s G F|p,(s) > 0}. Then U(t) is a neighborhood of t. By compactness of T \ U, there exist a finite number of points {/,, t2, . . ., tm) in T \ U such
that T\U
G UT U(t¡).Letp = (l/m)27/>,(. Thenp e 91,0 < p < 1,p(íq) = 0,
andp > Oon T\ U. Again using the compactness of T \ U, there exists 0 < 5 < 1 such thatp
> 8 on
T \ U. Let V = {/ G T\p(t) < 8/2). Then F is a neighborhood of t0 and V c U. Let k be the smallest integer which is greater than 1/5. Then k — 1 < 1/5 which
implies that k < (1 + 5)/5 < 2/5. Thus 1 < k8 < 2. Consider the functions q„ defined by
?n(0=[l
-/>"(')]*"
(«=1,2,...).
Clearly, q„ G 91,0 < qn < 1, and q„(t0) = 1. For each / G V, kp(t) < k8/2 < 1 so that, by Bernoulli's inequality,
qn(i)> l-[kp(t)]"> uniformly on V. For t G T\U,
l-(k8/2)"^l
kp(f) > k8 > 1 and, using Bernoulli's inequaüty,
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91
THE STONE-WEIERSTRASS THEOREM
qM = W(T)[l ~p"{t)] v(,) (1 — e/m)m > I — eonB. □ Finally, we turn to the proof of the Stone-Weierstrass theorem. Let/ G C(T) and e > 0. To complete the proof, it suffices to show the existence of a g G 91 such
that
|/(0 - g(t)\ 0. We may also assume that e ||/||. Define the sets Ay, Bj (j =
0, 1, . . . , n) by Aj={tG
T\f(t) < (/ -\)t),
Bj={tG
T\f(t) > (j+\)e).
Note that Aj and B, are disjoint closed sets in F, 0 = A0 c Ax G • ■ • G A„ = F, and Ä0 D 5, D • • • D 5„ = 0. For each 7 = 0, 1, . . . , n, Lemma 2 implies that there is x, G 91,with 0 < x¡ < 1, x¡ < e/« on 4,, and Xj > I — e/n on A. Then the function g = eS^ -*, is in 91. For any t G T, we have i e /^ \ A. , for somey > 1 which implies that
{j-\)e 2 J-2
g(t) > e 2 *,(') > U - l)e(l - e/n) o
= (j - l)e - ((j - l)/n)e2
> (j - l)e - e2 > {j - |)e.
The inequahty g(t) > (j — |)e is triviaUy true forj = 1. Thus
1/(0- *(0I< (J + !)« - (J - f)e < 2e. D References 1. H. Kuhn,
Ein elementarer
Beweis des Weierstrasschen
Approximationssatzes,
Arch, der Math.
(Basel) 15 (1964),316-317. Fachbereich
Mathematik, J. W. Goethe-Untversität
D-6000 Frankfurt,
Frankfurt,
Robert Mayer-Strasse
6-10,
West Germany (Current address of Bruno Brosowski)
Current address (Frank Deutsch): Department
of Mathematics,
versity Park, Pennsylvania 16802
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Pennsylvania State University, Uni-
