Prof. D. P.Patil, Department of Mathematics, Indian Institute of Science, Bangalore August-December 2004 MA-231 Topology 13. Function spaces1) —————————— — Uniform Convergence, Stone-Weierstrass theorem, Arzela-Ascoli theorem ———————————————————————————————– November 22, 2004 Karl Theodor Wilhelm Weierstrass† Marshall Harvey Stone†† (1815-1897) (1903-1989) First recall the following definitions and results : Our overall aim in this section is the study of the compactness and completeness properties of a subset F of the set Y X of all maps from a space X into a space Y . To do this a usable topology must be introduced on F (presumably related to the structures of X and Y ) and when this is has been done F is called a f unction space. N13.1 . (The Topology of Pointwise Convergence 2) Let X be any set, Y be any topological space and let fn : X → Y , n ∈ N, be a sequence of maps. We say that the sequence (fn)n∈N is pointwise convergent,ifforeverypoint x ∈X, the sequence fn(x) , n∈N,inY is convergent. a). The (Tychonoff) product topology on Y X is determined solely by the topology of Y (even if X is a topological X X space) the structure on X plays no part. A sequence fn : X →Y , n∈N,inY converges to a function f in Y if and only if for every point x ∈X, the sequence fn(x) , n∈N,inY is convergent. This provides the reason for the name the topology of pointwise convergence;this topology is also simply called the pointwise topology andisdenoted by Tptc . Suppose that Y is a Hausdorff topological space, then the map f = lim fn with f(x):= lim fn(x) n→∞ n→∞ is called the l i m i t m a p or the limit ofthesequence fn , n ∈ N. Note that for each x ∈ X, f(x)is uniquely determined, since Y is Hausdorff. b). (Compact subsets of in the topology of pointwise convergence) LetX be a set and let Y be a Hausdorff topological space and let F ⊆ Y X with the topology of pointwise convergence. Then F is compact if and only (i) F is pointwise closed, i.e. F is closed in the topology of pointwise convergence on X X Y . (ii) For each x ∈ X , πx (F) ={f(x) | f ∈ F } has a compact closure in Y , where πx : Y → Y is the x-th projection f → f(x). 1) The study of sets or spaces of functions began with the work of Ascoli in 1883 [“Le curve Limite di una Variet´a Data di Curve,” Mem. Accad. Lincei (3) 18, 521-586 (1883)], Arzela in 1889 [“Funzioni di Linee,” Atti della Reale Accademia dei Lincei, Rendiconti , 5, 342-348 (1889)] and Hadamard in 1898. These papers mark the beginning , not only of function space theory, but of general topology itself, for it was the questions which they raised that men like Frechet, Riesz, Weyl and finally Hausdorff were trying to answer. Coherent attempts to study topologies on spaces of functions in their own right began in 1935 with Tychonoff, who pointed out that his product topology on Y X is just the topology of pointwise convergence. The term function space is used much earlier in connection with questions of a topological nature about sets of functions. 2) The study of pointwise convergence of (sequences of) functions is as old as the calculus. The study of uniform convergence began hard on the heels of the formalization of the notion of limit by Cauchy. Cauchy in 1821 published a faulty proof of the false statement that the pointwise limit of a sequence of continuous functions is always continuous. Fourier and Abel found counter examples in the context of Fourier series. Dirichlet then analyzed Cauchy’s proof and found the mistake: the notion of pointwise convergence had to be replaced by uniform convergence. In the last half of the 19th century, in the hands of Heine, Weierstrass, Riemann and others, uniform convergence came into its own in applications to integrations theory and Fourier seris. D. P. Patil / Exercise Set 13 top04-e013 ; November 26, 2004 ; 9:45 a.m. 1 13.2 MA-231 Topology/ August-December 2004 13. Function Spaces c). Let X be a set and let Y be a Hausdorff topological space and let F ⊆ Y X. Let T be a topology on F .If T is compact and larger than Tptc, then the identity map id : (F, T) → (F, Tptc) is continuous and therefore a homeomorphism! (note that (F, Tptc) is Hasudorff, since Y is). compact topology N13.2 . (Uniform convergence) LetX be any set, (Y, d) be a metric space. Then in addition to the above pointwise convergence, we also define the uniform convergence : A sequence fn : X →Y , n∈N, of maps from X into Y converges uniformly toamapf : X →Y , if for every ε>0, there exits a n0 ∈N such that d f(x),fn(x) ≤ε for all n ≥ n0 and all x ∈X. a). The uniform convergence of fn : X →Y , n∈N, implies the pointwise convergence. Moreover, if Y is complete then we have the following Cauchy’s criterion for the uniform convergence : b). (Cauchy’s criterion for uniform convergence) Aseqeunce (fn) of functions fn : X →Y from a set X into a complete metric space Y converges uniformly if and only if for every ε>0, there exists a n0 ∈ N such that for all m, n≥n0 and all x ∈X,wehave d fm(x), fn(x) ≤ε. c). Let fn : X → Y , n ∈ N, be a uniformly convergent sequence of continuous maps from the topological space X into the metric space Y . Then the limit function f = lim fn is also continuous. ( Proof : Let a ∈ X ∈ N ≤ and ε>0 be given. Then there exits a n and a nhood U of a such that d f(x),f n(x) ε/3 for all ∈ ≤ ∈ ∈ ≤ + x X and d fn(a),f n(x) ε/3 for all x U. Then for x U,wehave d f(a),f(x) d f(a),fn(a) d fn(a), fn(x) + d fn(x), f (x) ≤ ε. d). Let fn : X → Y , n ∈ N,bealocally uniformly convergent sequence of continous maps from the topological space X in the metric space Y . Then the limit function lim fn is also continuous. (Hint : Immediate from the fact that continuity is a local property. — Recall that a sequence fn : X →Y locally uniformly convergent, if for every x ∈X, there exists a nhood U of x in X such that the sequence fn|U, n∈N, is uniformly convergent on U.) N13.3 . (The topology (or metric) of the uniform convergence on Y X) Let X be any set and let (Y, d) be a metric space. Then on the set Y X of all maps from X into Y , there is a natural metric defined by : ρ(f,g) := Sup Min d f(x),g(x) , 1 x ∈ X for f, g ∈ Y X. 3) Then : ρ is a metric Y X and the sequence X X X fn ∈Y , n∈N, uniformly converges to a map f ∈Y if and only if the sequence fn ∈Y , n∈N, in the metric space Y X converges to f . Therefore the topology of Y X defined by the metric ρ is called the topology (or metric) X of the uniform convergence on Y or just the u n i f o r m t opology andisdenoted by Tuc . a). Let X be a topological space and let Y be a metric space. Then the set C(X, Y ) of all continuous maps from X into Y is a closed subset of Y X. Further, if Y is complete then Y X is also complete and in particular, the set C(X, Y ) is a complete metric space with respect to the metric of uniform convergence. (Hint : Use 13.1-c) and the Cauchy’s criterion for uniform convergence 13.1-b)) b). If X is a compact topological space, then in this case we choose the natural distance between two continuous functions f, g : X → K asthe supremum norm or Tschebyschev norm defines a distance function g−f =g−f X = Sup |g(x)− f(x)| x ∈ X . Moreover, if X =∅ , then there exists a x0 ∈ X such that g−f X =|g(x0)−f(x0)| . In addition to the topology of pointwise convergence and the topology of uniform convergence, there are other interesting and useful topologies on Y X and C(X, Y ) , for example, the topology of compact convergence and the compact-open topology. We shall recall these below : N13.4 . (The topology of compact convergence on Y X) Let X be any topological space and let X (Y, d) be a metric space. For f ∈ Y , a real number ε>0 and a compact subset K of X, let BK (f ; ε) := { ∈ X | } { ; | ∈ X ⊆ } g Y supx∈K d (f (x), g(x)) <ε . The subsets BK (f ε) f Y ,ε >0 and a compact subset K X form a basis for a topology on Y X; this topology is called the topology of compact convergence or the topology of uniform convergence on compact sets. Thejustification for the choice of this X X terminology comes from the following : A sequence fn : X →Y , n∈N,inY converges to a function f in Y in K the topology of compact convergence if and only if for every compact subset K of X , the sequence fn|K ,n∈ Y converges uniformly to f |K in Y K .