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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 Ýmetric 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 2x  3y 
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 Ýmetric 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 Ýnorm, the solution of this exercise is identical 114 to that of part a with the Euclidean norm replaced by the Ýnorm.
c. Prove that in the metric space R k with the Ýmetric 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.
 Fall '08
 STAFF
 Math, Calculus

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