Analysis in Rn Math 203, Section 30 Autumn Quarter 2007 Paul Sally, e-mail: [email protected] John Boller, e-mail: [email protected] website: http://www.math.uchicago.edu/ boller/M203 ∼ Metric Spaces 0.1 Definition and Basic Properties of Metric Spaces Definition 0.1.1 A metric space is a pair (X, d) where X is a set and d : X X R is a map satisfying the following properties. × → a. For x , x X, d(x , x ) 0; 1 2 ∈ 1 2 ≥ we have d(x1, x2) = 0 if and only if x1 = x2, (positive definite). b. For any x , x X, we have d(x , x ) = d(x , x ), (symmetric). 1 2 ∈ 1 2 2 1 c. For any x , x , x X, we have 1 2 3 ∈ d(x , x ) d(x , x ) + d(x , x ), 1 2 ≤ 1 3 3 2 (triangle inequality). Exercise 0.1.2 i. Draw a triangle and figure out why the triangle inequality is so named. ii. Replace the triangle inequality by the inequality d(x , x ) d(x , x ) + d(x , x ) 1 2 ≤ 1 3 2 3 for any x1, x2, x3 X. Show that symmetry follows from this version of the triangle inequality and property a. ∈ n Exercise 0.1.3 On C = z = (z1, z2, . , zn) zj C , we define { | ∈ } 1/2 n z = z 2 ' ' | j| j=1 # and, for z, w Cn, we define d(z, w) = z w . Show that d is a metric on Cn. ∈ ' − ' Exercise 0.1.4 Let X be any nonempty set and, for x , x X, define 1 2 ∈ 0 if x = x d(x , x ) = 1 2 . 1 2 1 if x = x & 1 ) 2 Show that d is a metric on X. This is called the discrete metric. It is designed to disabuse people of the notion that every metric looks like the usual metric on Rn. The discrete metric is very handy for producing counterexamples. Definition 0.1.5 We introduce an important collection of metrics on Rn. n Let p be a real number such that p 1. For x = (x1, x2, . , xn) R , we define ≥ ∈ 1/p n x = x p . ' 'p | j| j=1 # n n As usual, if x = (x1, x2, . , xn) R and y = (y1, y2, . , yn) R , we define dp(x, y) = x y p. ∈ ∈ ' − ' Definition 0.1.6 If I R is an interval, the function f : I R is said to be convex on I provided that, given any λ [0, 1],⊂we have f(λx + (1 λ)y) λf(x) +→(1 λ)f(y). ∈ − ≤ − Lemma 0.1.7 The function f(x) = ex is convex on R. Theorem 0.1.8 (H¨older’s Inequality) Suppose p, q are real numbers greater than 1 such that 1/p + n n 1/q = 1. Suppose x = (x1, x2, . , xn) R and y = (y1, y2, . , yn) R , then ∈ ∈ n n 1/p n 1/q p q xkyk xk yk | | ≤ ' | | ( ' | | ( k#=1 k#=1 k#=1 (Hint: Let x = es/p and y = et/q.) | i| | i| n Exercise 0.1.9 Now prove that dp is a metric on R . p (Hint: To prove the Triangle Inequality, use H¨older’s Inequality and the observation that (xi +yi) = p 1 p 1 xi(xi + yi) − + yi(xi + yi) − .) Exercise 0.1.10 Note that H¨older’s inequality works for p, q > 1. Prove the triangle inequality for the d1 metric. Definition 0.1.11 We also define a metric for p = . That is, if x = (x , x , . , x ), we set ∞ 1 2 n x = max1 j n xj , and define d (x, y) = max1 j n xj yj = x y . ' '∞ ≤ ≤ | | ∞ ≤ ≤ | − | ' − '∞ Exercise 0.1.12 Prove that d defines a metric on Rn. ∞ n n p Definition 0.1.13 The space (R , dp) or alternatively (R , p), 1 p , is denoted by "n(R). 'n· ' ≤ ≤ ∞ Note that, in our present notation, the norm symbol on R should be relabeled 2. ' · ' ' · ' Exercise 0.1.14 Show that everything we have just done for Rn can also be done for Cn. This yields p a collection of spaces "n(C). 0.2 Topology of metric spaces Definition 0.2.1 Suppose that (X, d) is a metric space and x0 X. If r R, with r > 0, the open ball of radius r around x is the subset of X defined by B (x ) =∈x X d∈(x, x ) < r . 0 r 0 { ∈ | 0 } Exercise 0.2.2 In R2, with the usual metric, illustrate a ball of radius 5/2 around the point ( 1, 4). − 2 Exercise 0.2.3 In R , illustrate a ball of radius 5/2 around the point ( 1, 4) in the d1 metric. − Definition 0.2.4 Suppose that V is a vector space with a metric d. The unit ball in V is the ball of radius 1 with center at 0, that is B1(0). p n Definition 0.2.5 The unit ball in " (R) is the set of all points x R such that x p < 1. n ∈ ' ' 1 2 Exercise 0.2.6 For n = 2, illustrate the unit balls in "2(R), "2(R), and "2∞(R). Exercise 0.2.7 If 1 p < q, show that the unit ball in "p (R) is contained in the unit ball in "q (R). ≤ n n 2 1 Exercise 0.2.8 Consider the set of all points in R which lie outside the unit ball in "2(R) and inside p the unit ball in "2∞(R). Does every point in this region lie on the perimeter of the unit ball in "2(R) for some p between 1 and ? Do the same problem for "p (R). ∞ n Definition 0.2.9 Let (X, d) be a metric space and suppose that A X. The set A is an open set in X if, for each a A, there is an r > 0 such that B (a) A. ⊆ ∈ r ⊆ Exercise 0.2.10 Show that the empty set and the whole space X are both open sets. ∅ Exercise 0.2.11 Prove that, for any x0 X and any r > 0, the “open ball” Br(x0) is open. So now we can legitimately call an “open” ball an∈open set. Exercise 0.2.12 Prove that the following are open sets: i. The “first quadrant,” (x, y) R2 x > 0 and y > 0 , in the usual metric; { ∈ | } ii. any subset of of a discrete metric space. Theorem 0.2.13 i. If Aj j J is a family of open sets in a metric space (X, d), then { } ∈ Aj j J )∈ is an open set in X; ii. if A1, A2, . , An are open sets in a metric space (X, d), then n Aj j=1 * is an open set in X. Exercise 0.2.14 2 i. There can be problems with infinite intersections. For example, let An = B1/n((0, 0)) in R with the usual metric. Show that ∞ An n=1 * is not open. ii. Find an infinite collection of distinct open sets in R2 with the usual metric whose intersection is a nonempty open set. Definition 0.2.15 Let (X, d) be a metric space and suppose that A X. We say that A is a closed set in X if cA is open in X. (Recall that cA = X A is the complemen⊆t of A in X.) \ Exercise 0.2.16 Show that the following are closed sets. i. The x-axis in R2 with the usual metric; ii. the whole space X in any metric space; iii. the empty set in any metric space; iv. a single point in any metric space; v. any subset of a discrete metric space. Exercise 0.2.17 Show that Q as a subset of R with the usual metric is neither open nor closed in R. On the other hand, show that if the metric space is simply Q with the usual metric, then Q is both open and closed in Q. Theorem 0.2.18 i. Suppose that (X, d) is a metric space and that Aj j J is a collection of closed sets in X. Then { } ∈ Aj j J *∈ is a closed set in X; ii. if A1, A2, . , An are closed sets in X, then n Aj j=1 ) is a closed set in X. Definition 0.2.19 Suppose that A is a subset of a metric space X. A point x0 X is an accumulation point of A if, for every r > 0, we have (B (x ) x ) A = . ∈ r 0 \ { 0} ∩ ) ∅ Exercise 0.2.20 Give an example of a metric X and a set A X that has at least one accumulation point in A as well as at least one accumulation point not in A.⊆ Definition 0.2.21 Suppose that A is a subset of a metric space X. A point x0 A is an isolated point of A if there is an r > 0 such that B (x ) A = x . ∈ r 0 ∩ { 0} Definition 0.2.22 Suppose that A is a subset of a metric space X. A point x0 X is a boundary c ∈ point of A if, for every r > 0, Br(x0) A = and Br(x0) A = . The boundary of A is the set of boundary points of A, and is denoted b∩y ∂A) .∅ ∩ ) ∅ Definition 0.2.23 i. Let A = (x, y, z) R3 x2 + y2 + z2 < 1 .