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Complete Metric Space Complete Metric Space The aim of this chapter is to study one of the properties of metric space. The notion of distance between points of an abstract set leads naturally to the discussion of convergence of sequences and Cauchy sequences in the set. Unlike the situation of real numbers, where each Cauchy sequence is convergent, there are metric spaces in which Cauchy sequences fail to converge. A metric space in which every Cauchy sequence converges is called a `complete metric space'. This property plays a vital role in analysis when one wishes to make an existence statement. We shall see that a metric space need not be complete and hence we shall find conditions under which such a property can be ensured. 1 Sequence Let (X; d) be a metric space. A sequence of points in X is a function x : N ! X: In other words, a sequence assigns to each n 2 N a uniquely determined element of X: If f(n) = xn; it is customary to denote the sequence by the symbol hxni; or by x1; x2; ··· ; xn; ··· : (i) Let hxni be a sequence in the set X. If there exists n0 2 N such that xn = a for all n > n0, then the sequence hxni is called eventually constant. (ii) If xn = n for all n 2 N, then the sequence hxni is called constant sequence. Obviously, a constant sequence is a special case (n0 = 1) of an eventually constant sequence. (iii) A sequence hxni is said be frequently in a set S if for each positive integer m 2 N, there exists a positive integer n ≥ m such that xn 2 S. (iv) Suppose X is a non-empty set and x = hxni is a sequence in X. For each m 2 N, the set th fxn : n 2 N; n ≥ :mg is called the m tail of the sequence hxni: 1.1 Convergent Sequence The concept of a convergent sequence plays an important role to investigate the closedness of a set, the continuity of a function and several other properties. In this section, we give an introduction to the convergence of sequences and the properties under which a sequence is convergent in arbitrary metric spaces. Definition 1 [Convergence of a Sequence] Let (X; d) be a metric space and A be a non- empty subset of X. A sequence hxni in X is said to be convergent if there is a point a 2 X such that for each real " > 0, 9 a k 2 N such that d(xn; a) < " 8 n ≥ k. Equivalently for each open sphere S(a; ") centred at a, 9 a positive integer k such that xn 2 S(a; "); 8n ≥ k: (1) Here a is called the limit of the sequence hxni and we write lim xn = a. n!1 1 2 Small Overview On Complete Metric Space (i) Therefore a sequence hxni in a matric space (X; d) converges to a if and only if the hd(xn; a)ii in Ru converges to zero. (ii) A singleton set in a metric space is, of course, included in every ball centred at its only point. A sequence that has a singleton set for a tail is said to be eventually constant; such sequences must converge in any metric space to which they belong. Thus, an eventually constant sequence, and hence a constant sequence, is convergent. (iii) A constant sequence, in which all terms are the same, is a special case of an eventually constant sequence and converges to its single value in any metric space to which it belongs. (iv) In a discrete metric space X; it is very difficult for a sequence to converge. Let (X; d) be a discrete metric space, and hxni be a sequence in X, which converges to a in X. Let " = 1=2: Since hxni converges to a, 9m 2 N such that d(xn; ) < " = 1=2 for all n ≥ m: This means that xn = a for n ≥ m, i.e., the sequence is of the form hx1; x2; ··· ; xm−1; a; a; · · · i. Each singleton set fag is open and the only way that fag can include a tail of a sequence hxni is if the sequence is eventually constant with xn = a for all sufficiently large n 2 N: Thus in a discrete metric space, a sequence can converge to a point only if it is an eventually constant sequence. (v) The sequence h1=ni of inverses of the natural numbers converges to 0 because for each + th r 2 R there exists k 2 N such that 1=k < r and then the ball B[0; r) includes the k tail of the sequence h1=ni. (vi) Let X denote the space of all sequences of numbers with metric d defined by 1 X 1 jxj − yjj d(x; y) = ; x = hx i; y = hy i 2 X: 2j 1 + jx − y j j j j=1 j j (n) (n) Let hx i = hhxj ii be a sequence in X which converges to, say, x 2 X: In other words, 1 (n) X 1 jxj − xjj d(x(n); x) = ! 0 as n ! 1: 2j (n) j=1 1 + jxj − xjj This means that, for " > 0; there exists an integer n0(") 2 N such that 1 (n) X 1 jxj − xjj < "; whenever n ≥ n (") 2j (n) 0 j=1 1 + jxj − xjj (n) 1 jx − xjj ) j < "; whenever n ≥ n ("); j = 1; 2; ··· : 2j (n) 0 1 + jxj − xjj (n) Since " > 0 is arbitrary, it follows that lim x = xj for each j. Consider any " > 0: n!1 j 1 There exists an integer m such that P 1=2j < "=2. Consequently, j=m+1 m (n) 1 (n) X 1 jxj − xjj X 1 jxj − xjj d x(n); x = + 2j (n) 2j (n) j=1 1 + jxj − xjj j=m+1 1 + jxj − xjj m (n) X 1 jxj − xjj " < + : 2j (n) 2 j=1 1 + jxj − xjj Dr.Prasun Kumar Nayak & Dr.Himadri Shekhar Mondol Home Study Materiel 3 Small Overview On Complete Metric Space (n) As the first sum contains only finitely many terms and lim x = xj for each j, there n!1 j exists n0(") 2 N such that m (n) X 1 jxj − xjj < "=2; whenever n ≥ n ("): 2j (n) 0 j=1 1 + jxj − xjj (n) Hence, d(x ; x) < " whenever n ≥ n0("). Thus, convergence in the space of all sequences is co-ordinatewise convergence. The metric space is known as Frechet space. Result 1 (Criteria for Convergence): Suppose X is a metric space, a 2 X and hxni is a sequence in X. Then following are some equivalent statements, any one of which might be used in a definition of convergence in metric spaces: \ DD E E (i) (closure criterion I): xn : n 2 S j S ⊆ N;S = infinite = hai: \ DD E E (ii) (closure criterion II): x 2 xn : n 2 S j S ⊆ N;S = infinite : (iii) (distance criterion) : dist a; fxn : n 2 Sg = 0 for every infinite subset S of N. (iv) (ball criterion): Every open ball centred at a includes a tail of hxni: (v) (open set criterion): Every open subset of X that contains a includes a tail of hxni: Theorem 1 In a metric space, every convergent sequence has an unique limit. Proof: Let (X; d) be a metric space and hxni be a convergent sequence in X. If possible, let a and b be two limits of hxni, i.e., a 6= b, so d(a; b) > 0. Since lim xn = a and lim xn = b, so n!1 n!1 corresponding to an " > 0, 9 n1; n2 2 N such that d(xn; a) < "=2; for n ≥ n1 and d(xn; b) < "=2; for n ≥ n2: Using the triangle inequality, we have d(xn; xn) ≤ d(xn; a) + d(a; b) + d(xn; b) or , d(xn; xn) − d(a; b) ≤ d(xn; b) + d(xn; a): (2) Also, interchanging xn; a and xn; b in Eq.(2), we obtain d(a; b) − d(xn; xn) ≤ d(xn; b) + d(xn; a): (3) Combining (2) and (3), we get h i h i − d(xn; b) + d(a; xn) ≤ − d(xn; xn) − d(a; b) ≤ d(xn; b) + d(xn; a) ) d(xn; xn) − d(a; b) ≤ d(xn; b) + d(xn; a) ) 0 ≤ d(xn; xn) − d(a; b) ≤ "; for n ≥ m = maxfn1; n2g ) 0 = d(xn; xn) ! d(a; b): ) d(a; b) = 0 and so a = b. Thus the limit of the sequence hxni is unique. Note: A sequence hxni which is not convergent is called divergent. Dr.Prasun Kumar Nayak & Dr.Himadri Shekhar Mondol Home Study Materiel 4 Small Overview On Complete Metric Space 1.2 Convergence of Subsequences Every subsequence of a convergent sequence converges to the same limit as the parent sequence. A sequence that does not converge, however, may have many convergent subsequences with various limits. We discover in this section what those limits are. Definition 2 Suppose X is a metric space, x 2 X and hxni is a sequence in X that converges to x. Suppose hxmn i is a subsequence of hxni. Then the sequence hxmn i also converges to x. Theorem 2 Let hxni be a convergent sequence in a metric space (X; d) such that xn ! x as n ! 1. If hxnk i is any subsequence of hxni then xnk ! a as k ! 1: Proof: Since hxni converges to a for any " > 0; 9 a natural number m such that d(xn; a) < "; 8 n ≥ m: Since hnki is a strictly monotonically sequence of natural number so 9 a k1 2 N such that nk ≥ m; 8 k ≥ k1: Form these, we get d(xnk ; a) ≤ d(xnk ; xn) + d(xn; a) < "; 8 k ≥ k1: ) The sub-sequence hxnk i converges to a: Thus every subsequence of a convergent sequence is convergent.
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