Chapter 3 METRIC SPACES and SOME BASIC TOPOLOGY Thus far, our focus has been on studying, reviewing, and/or developing an under- standing and ability to make use of properties of U U1. The next goal is to generalize our work to Un and, eventually, to study functions on Un. 3.1 Euclidean n-space The set Un is an extension of the concept of the Cartesian product of two sets that was studied in MAT108. For completeness, we include the following De¿nition 3.1.1 Let S and T be sets. The Cartesian product of S and T , denoted by S T,is p q : p + S F q + T . The Cartesian product of any ¿nite number of sets S1 S2 SN , denoted by S1 S2 SN ,is j b ck p1 p2 pN : 1 j j + M F 1 n j n N " p j + S j . The object p1 p2pN is called an N-tuple. Our primary interest is going to be the case where each set is the set of real numbers. 73 74 CHAPTER 3. METRIC SPACES AND SOME BASIC TOPOLOGY De¿nition 3.1.2 Real n-space,denotedUn, is the set all ordered n-tuples of real numbers i.e., n U x1 x2 xn : x1 x2 xn + U . Un U U U U Thus, _ ^] `, the Cartesian product of with itself n times. nofthem Remark 3.1.3 From MAT108, recall the de¿nition of an ordered pair: a b a a b . def This de¿nition leads to the more familiar statement that a b c d if and only if a bandc d. It also follows from the de¿nition that, for sets A, B and C, A B C is, in general, not equal to A B C i.e., the Cartesian product is not associative. Hence, some conventions are introduced in order to give meaning to the extension of the binary operation to more that two sets. If we de¿ne ordered triples in terms of ordered pairs by setting a b c a b c this would allow us to claim that a b c x y z if and only if a x, b y, and c z. With this in mind, we interpret the Cartesian product of sets that are themselves Cartesian products as “big” Cartesian products with each entry in the tuple inheriting restrictions from the original sets. The point is to have helpful descriptions of objects that are described in terms of n-tuple. Addition and scalar multiplication on n-tuple is de¿ned by x1 x2xn y1 y2 yn x1 y1 x2 y2xn yn and : x1 x2 xn :x1:x2 :xn ,for: + U, respectively. The geometric meaning of addition and scalar multiplication over U2 and U3 as well as other properties of these vector spaces was the subject of extensive study in vector calculus courses (MAT21D on this campus). For each n, n o 2, it can be shown that Un is a real vector space. De¿nition 3.1.4 A real vector space Y is a set of elements called vectors,with given operations of vector addition : Y Y Y and scalar multiplication : U Y Y that satisfy each of the following: 3.1. EUCLIDEAN N-SPACE 75 1. 1v1wv w + Y " v w w v commutativity 2. 1u1v1wu v w + Y " u v w u v w associativity 3. 200 + Y F 1vv + Y " 0 v v 0 v zero vector 4. 1vv + Y " 2 v v + Y F v v v v 0 negatives 5. 1D1v1wD + U F v w + Y " D v w D v D w distribu- tivity 6. 1D1< 1wD < + U F w + Y " D < w D< w associativity 7. 1D 1<1wD< + U F w + Y " D < w D w< w distribu- tivity 8. 1vv + Y " 1 v v 1 v multiplicative identity n Given two vectors, x x1 x2xn and y y1 y2 yn in U ,theinner product (also known as the scalar product) is ;n x y x j y j j1 n and the Euclidean norm (ormagnitude)ofx x1 x2 xn + U is given by Y X T X;n b c W 2 x x x x j . j1 The vector space Un together with the inner product and Euclidean norm is called Euclidean n-space. The following two theorems pull together the basic properties that are satis¿ed by the Euclidean norm. Theorem 3.1.5 Suppose that x y z + Un and : + U. Then (a) x o 0 (b) x 0 % x 0 (c) :x : x and 76 CHAPTER 3. METRIC SPACES AND SOME BASIC TOPOLOGY (d) x y n x y . Excursion 3.1.6 Use Schwarz’s Inequality to justify part (d). For x x1 x2 xn n and y y1 y2 yn in U , x y 2 Remark 3.1.7 It often helps to take our observations back to the setting that is U1 U2 “once removed” from . For the case , the statement given in part (d) of the theorem relates to the dot product of two vectors: For G x1 x2 and @ y1 y2, we have that G @ x1 y1 x2 y2 which, in vector calculus, was shown to be equivalent to G @ cosA where A is the angle between the vectors G and @. Theorem 3.1.8 (The Triangular Inequalities) Suppose that x x1 x2 xN , N y y1 y2yN and z z1 z2 zN are elements of U . Then (a) x y n x y i.e., ;N 1 2 ;N 1 2 ;N 1 2 2 n 2 2 x j y j x j y j j1 j1 j1 where 12 denotes the positive square root and equality holds if and only if either all the x j are zero or there is a nonnegative real number D such that y j Dx j for each j, 1 n j n N and (b) x z n x y y z i.e., ;N 1 2 ;N 1 2 ;N 1 2 2 2 2 x j z j n x j y j y j z j j1 j1 j1 3.2. METRIC SPACES 77 where 12 denotes the positive square root and equality holds if and only if there is a real number r, with 0 n r n 1, such that y j rxj 1 rz j for each j, 1 n j n N. Remark 3.1.9 Again, it is useful to view the triangular inequalities on “familiar ground.” Let G x1 x2 and @ y1 y2. Then the inequalities given in The- orem 3.1.8 correspond to the statements that were given for the complex numbers i.e., statements concerning the lengths of the vectors that form the triangles that are associated with ¿nding G @ and G @. Observe that, for C x y : x2 y2 1 and I x : a n x n b where a b, the Cartesian product of the circle C with I, C I, is the right circular cylinder, U x y z : x2 y2 1 F a n z n b and the Cartesian product of I with C, I C, is the right circular cylinder, V x y z : a n x n b, y2 z2 1 If graphed on the same U3-coordinate system, U and V are different objects due to different orientation on the other hand, U and V have the same height and radius which yield the same volume, surface area etc. Consequently, distinguishing U from V depends on perspective and reason for study. In the next section, we lay the foundation for properties that place U and V in the same category. 3.2 Metric Spaces In the study of U1 and functions on U1 the length of intervals and intervals to de- scribe set properties are useful tools. Our starting point for describing properties for sets in Un is with a formulation of a generalization of distance. It should come as no surprise that the generalization leads us to multiple interpretations. De¿nition 3.2.1 Let S be a set and suppose that d : S S U1.Thendissaid to be a metric (distance function) on S if and only if it satis¿es the following three properties: d e (i) 1x1y x y + S S " dx y o 0 F dx y 0 % x y , 78 CHAPTER 3. METRIC SPACES AND SOME BASIC TOPOLOGY d e (ii) 1x1y x y + S S " dy x dx y (symmetry), and d e (iii) 1x1y1z x y z + S " dx z n dx y dy z (triangle inequal- ity). De¿nition 3.2.2 A metric space consists of a pair S d–a set, S, and a metric, d, on S. Remark 3.2.3 There are three commonly used (studied) metrics for the set UN . For x x1 x2xN and y y1 y2 yN , we have: T3 b c UN N 2 d where d x y j1 x j y j , the Euclidean metric, 3 UN N D where D x y j1 x j y j , and n n N n n U d* where d*x y max x j y j . 1n jnN Proving that d, D,andd* are metrics is left as an exercise. Excursion 3.2.4 Graph each of the following on Cartesian coordinate systems 1. A x + U2 : d0 x n 1 2. B x + U2 : D0 x n 1 3.2. METRIC SPACES 79 3. C x + U2 : d*0 x n 1 ***For (1), you should have gotten the closed circle with center at origin and radius one for (2), your work should have led you to a “diamond” having vertices at 1 0, 0 1, 1 0,and0 1 the closed shape for (3) is the square with vertices 1 1, 1 1, 1 1,and1 1.*** Though we haven’t de¿ned continuous and integrable functions yet as a part of this course, we offer the following observation to make the point that metric spaces can be over different objects.