1873_solutions

B c prove that if a is any nonempty subset of r k

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Unformatted text preview: s defined in Exercises 2 and 3. Of course, with the Euclidean metric, the ball is the disk x 2  y 2 1. 1, 0 With the Ý-metric the condition for a point x, y to lie in the ball is that the larger of |x | and |y | does not exceed 1. In other words, x, y B O, 1 if and only if |x | 1 and |y | 1. So the ball is a square. 1, 1 1, 1 1, 1 1, 1 With the metric of Exercise 2, the condition x, y xy  1 x y1 113 B O, 1 says that |x |  |y | xy  1 x y1 1. Finally, with the metric of Exercise 3, the ball is the set of points x, y for which 2|x |  3|y | 2x  3y  1 2x 1 2x  3y  1 2x 3y  1 3y  1 5. Prove that a metric space X is bounded if and only if it is possible to find a positive number r and a member c of the space X such that X  B c, r . If X  B c, r and x and y belong to X then d x, y d x, c  d c, y r  r  2r which tells us that diam X 2r. On the other hand, if we know that X is bounded and choose r  0 such that d x, y  r whenever x and y belong to X then, given any point c X we have X  B c, r . 6. Prove that a metric space X is bounded if and only if it is possible to find a positive number r such that for every member c of the space X we have X  B c, r . The solution given to Exercise 5 satisfies the conditions stated in this exercise too. 7. Suppose that x and y are points in a metric space X and that x   d x, y . Prove that B x,  B y,  2 2 Solution: Given any member u of the ball B  2  2 . we have d x, u  d u, y d x, y and since d x, u  x, y. Suppose that we obtain   d u, y 2  . Therefore no member of the ball B x, 2  from which we conclude that d u, y ball B y,  . 2 8.  2 can belong to the a. Prove that in the metric space R k with the Euclidean metric the diameter of every ball B c, r and every ball B c, r is 2r. On the one hand we know that whenever x and y belong to the ball B c, r we have xy x cc y r  r  2r and we conclude that the diameter of B c, r does not exceed 2r. Now, to show that the diameter of B c, r can’t be less than 2r, suppose that 0  p  2r. We shall find two points x and y in the ball B c, r such that x y  2r. We begin by choosing a number q between p and 2r. Then we define q x  c e 2 and q e yc 2 and observe that x and y belong to B c, r and that x y  q  p. b. Prove that in the metric space R k with the Ý-metric the diameter of every cube I c, r and every cube I c, r is 2r. Since the cubes are just the balls in R k with the Ý-norm, the solution of this exercise is identical 114 to that of part a with the Euclidean norm replaced by the Ý-norm. c. Prove that in the metric space R k with the Ý-metric the diameter of every ball B c, r and every ball B c, r is 2r. Given any points x and y in B c, r we have x yÝ x y  2r. on the other hand, if we follow the argument given in the solution of part a then we obtain q q x y Ý c e c e  qe Ý  qe  q 2 2 Ý and so the argument given there applies here too. d. Prove that in the metric space R k with the Euclidean metric the diameter of...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.

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