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Spaces of Functions CHAPTER 8 Spaces of functions 1. The algebra (X) of continuous functions C 2. Approximations in (X): the Stone-Weierstrass theorem C 3. Recovering X from (X): the Gelfand Naimark theorem C 4. General function spaces (X,Y ) C Pointwise convergence, uniform convergence, compact convergence • Equicontinuity • Boundedness • The case when X is a compact metric spaces • The Arzela-Ascoli theorem • The compact-open topology • 5. More exercises 105 106 8. SPACES OF FUNCTIONS 1. The algebra (X) of continuous functions C We start this chapter with a discussion of continuous functions from a Hausdorff compact space to the real or complex numbers. It makes no difference whether we work over R or C, so let’s just use the notation K for one of these base fields and we call it the field of scalars. For each z K, we can talk about z R- the absolute value of z in the real case, or the norm of the complex∈ number z in the complex| | ∈ case. For a compact Hausdorff space X, we consider the set of scalar-valued functions on X: (X) := f : X K : f is continuous . C { → } When we want to make a distinction between the real and complex case, we will use the more precise notations (X, R) and (X, C). In this sectionC we look closerC at the “structure” that is present on (X). First, there is a topological one. When X was an interval in R, this was discussed in SectionC 9, Chapter 3 (there n = 1 in the real case, n = 2 in the complex one). As there, there is a metric on (X): C d (f, g) := sup f(x) g(x) : x R . sup {| − | ∈ } Since f, g are continuous and X is compact, dsup(f, g) < . As in loc.cit., dsup is a metric and the induced topology is called the uniform topology on (∞X). And, still as in loc.cit.: C Theorem 8.1. For any compact Hausdorff space X, ( (X), dsup) is a complete metric space. C Proof. Let (fn) be a Cauchy sequence in (X). The proof of Theorem 3.27 applies word by C word to our X instead of the interval I, to obtain a function f : X R such that dsup(fn,f) 0 when n . Then similarly, the proof of Theorem 3.26 (namely→ that (I, Rn) is closed→ in → ∞ C ( (I, Rn), dˆ )) applies word by word with I replaced by X to conclude that f (X). F sup ∈C The metric on (X) is of a special type: it comes from a norm. Namely, defining C f := sup f(x) : x X [0, ) || ||sup {| | ∈ }∈ ∞ for f (X), we have ∈C d (f, g)= f g . sup || − ||sup Definition 8.2. Let V be a vector space (over our K = R or C). A norm on V is a function : V [0, ), v v || · || → ∞ 7→ || || such that v = 0 v = 0 || || ⇐⇒ and is compatible with the vector space structure in the sense that: λv = λ v λ K, v V, || || | | · || || ∀ ∈ ∈ v + w v + w v, w V. || || ≤ || || || || ∀ ∈ The metric associated to is the metric d given by || · || ||·|| d (v, w) := v w . ||·|| || − || A Banach space is a vector space V endowed with a norm such that d||·|| is complete. When K = R we talk about real Banach spaces, when K = C about|| · || complex ones. With these, we can now reformulate our discussion as follows: Corollary 8.3. For any compact Hausdorff space X, ( (X), sup) is a Banach space. C || · || 1. THE ALGEBRA C(X)OFCONTINUOUSFUNCTIONS 107 Of course, this already makes reference to some of the algebraic structure on (X)- that of vector space. But, of course, there is one more natural operation on continuous functions:C the multiplication, defined pointwise: (fg)(x)= f(x)g(x). Definition 8.4. A K-algebra is a vector space A over K together with an operation A A A, (a, b) a b × → 7→ · which is unital in the sense that there exists an element 1 A such that ∈ 1 a = a 1= a a A, · · ∀ ∈ and which is K-bilinear and associative, i.e., for all a, a′, b, b′, c A, λ K, ∈ ∈ (a + a′) b = a b + a′ b, a (b + b′)= a b + a b′, · · · · · · (λa) b = λ(a b)= a (λb), , · · · a (b c) = (a b) c . · · · · We say that A is commutative if a b = b a for all a, b . · · ∈A Example 8.5. The space of polynomials K[X1,...,Xn] in n variables (with coefficients in K) is a commutative algebra. Hence the algebraic structure of (X) is that of an algebra. Of course, the algebraic and the topological structures are compatible.C Here is the precise abstract definition. Definition 8.6. A Banach algebra (over K) is an algebra A equipped with a norm which makes (A, ) into a Banach space, such that the algebra structure and the norm are compatible,|| · || in the sense||·|| that, a b a b a, b A. || · || ≤ || || · || || ∀ ∈ Hence, with all these terminology, the full structure of (X) is summarized in the following: C Corollary 8.7. For any compact Hausdorff space, (X) is a Banach algebra. C There is a bit more one can say in the case when K = C: there is also the operation of conjugation, defined again pointwise: f(x) := f(x). As before, this comes with an abstract definition: Definition 8.8. A *-algebra is an algebra A over C together with an operation ( )∗ : A A, a a∗, − → 7→ which is an involution, i.e. (a∗)∗ = a a A, ∀ ∈ and which satisfies the following compatibility relations with the rest of the structure: (λa)∗ = λa∗ a A, λ C, ∀ ∈ ∈ 1∗ = 1, (a + b)∗ = a∗ + b∗, (ab)∗ = b∗a∗ a, b A. ∀ ∈ Finally, a C∗-algebra is a Banach algebra (A, ) endowed with a *-algebra structure, s.t. || · || a∗ = a , a∗a = a 2 a A. || || || || || || || || ∀ ∈ Of course, C with its norm is the simplest example of C∗-algebra. To summarize our discussion in the complex case: Corollary 8.9. For any compact Hausdorff space, (X, C) is a C∗-algebra. C 108 8. SPACES OF FUNCTIONS 2. Approximations in (X): the Stone-Weierstrass theorem C The Stone-Weierstrass theorem is concerned with density in the space (X) (endowed with the uniform topology); the simplest example is Weierstrass’s approximationC theorem which says that, when X is a compact interval, the set of polynomial functions is dense in the space of all continuous functions. The general criterion makes use of the algebraic structure on (X). C Definition 8.10. Given an algebra A (over the base field R or C), a subalgebra is any vector subspace B A, containing the unit 1 and such that ⊂ b b′ B b, b′ B. · ∈ ∀ ∈ When we want to be more specific about the base field, we talk about real or complex subalgebras. Example 8.11. When X = [0, 1], the set of polynomial functions on [0, 1] is a unital subal- gebra of ([0, 1]). Here the base field can be either R or C. C Definition 8.12. Given a topological space X and a subset (X), we say that is point-separating if for any x,y X, x = y there exists f suchA⊂C that f(x) = f(y). A ∈ 6 ∈A 6 Example 8.13. When X = [0, 1], the subalgebra of polynomial functions is point-separating. Here is the Stone-Weierstrass theorem in the real case (K = R). Theorem 8.14. (Stone-Weierstrass) Let X be a compact Hausdorff space. Then any point- separating real subalgebra (X, R) is dense in ( (X, R), dsup). A⊂C C Proof. We first show that there exists a sequence (pn)n≥1 of real polynomials which, on the interval [0, 1], converges uniformly to the function √t. We construct pn inductively by 1 2 pn (t)= pn(t)+ (t pn(t) ), p = 0. +1 2 − 1 We first claim that pn(t) √t for all t [0, 1]. This follows by induction on n since ≤ ∈ √t + pn(t) √t pn (t) = (√t pn(t))(1 ). − +1 − − 2 and pn(t) √t implies that √t + pn(t) 2√t 2 for all t [0, 1] (hence the right hand side ≤ ≤ ≤ ∈ is positive). Next, the recurrence relation implies that pn (t) pn(t) for all t. Then, for each +1 ≥ t [0, 1], (pn(t))n is increasing and bounded above by 1, hence convergent; let p(t) be its ∈ ≥1 limit. By passing to the limit in the recurrence relation we find that p(t)= √t. We still have to show that pn converges uniformly to p on [0, 1]. Let ǫ > 0 and we look for N such that p(t) pn(t) < ǫ for all n N (note that p pn is positive). Let t [0, 1]. Since − ≥ − ∈ pn(t) p(t), we find N(t) such that p(t) pn(t) < ǫ/3 for all n N(t). Since p and p are → − ≥ N(t) continuous, we find an open neighborhood V (t) of t such that p(s) p(t) < ǫ/3 and similarly for p , for all s V (t). Note that, for each s V (t) we have| the desired− | inequality: N(t) ∈ ∈ ǫ p(s) p (s) = (p(s) p(t)) + (p(t) p (t)) + (p (t) p (s)) < 3 = ǫ. − N(t) − − N(t) N(t) − N(t) 3 Varying t, V (t) : t [0, 1] will be an open cover of [0, 1] hence we can extract an open sub- { ∈ } cover V (t ),...,V (tk) .
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