The Real And Complex Number Systems

point of M. Sincx is arbitrary, we know that every point of M is interior. So, M is open.
Let x be an adherent point of M, it is clearly x M since we consider all points lie in
M. Hence, M contains its all adherent points. It implies that M is closed.
Remark: The reader should regard the statement as a common sense.
3.27 Consider the following two metrics in R n :
in
d 1 x, y max 1in |x i y i |, d 2 x, y |xi i1
y i |.
In each of the following metric spaces prove that the ball Ba; r has the geometric
appearance indicated:
(a) In R 2 , d 1 , a square with sides parallel to the coordinate axes.
Solution: It suffices to consider the case B0, 0,1. Letx x 1 , x 2 B0, 0,1,
then we have
|x 1 | 1, and |x 2 | 1.
So, it means that the ball B0, 0,1 is a square with sides lying on the coordinate axes.
Hence, we know that Ba; r is a square with sides parallel to the coordinate axes.
(b) In R 2 , d 2 , a square with diagonals parallel to the axes.
Solution: It suffices to consider the case B0, 0,1. Letx x 1 , x 2 B0, 0,1,
then we have
|x 1 x 2 | 1.
So, it means that the ball B0, 0,1 is a square with diagonals lying on the coordinate
axes. Hence, we know that Ba; r is a square with diagonals parallel to the coordinate
axes.
(c) A cube in R 3 , d 1 .
Solution:It suffices to consider the case B0, 0, 0,1. Let
x x 1 , x 2 , x 3 B0, 0, 0,1, then we have
|x 1 | 1, |x 2 | 1, and |x 3 | 1.
So, it means that the ball B0, 0, 0,1 is a cube with length 2. Hence, we know that
Ba; r is a cube with length 2a.
(d) An octahedron in R 3 , d 2 .
Solution: It suffices to consider the case B0, 0, 0,1. Let
x x 1 , x 2 , x 3 B0, 0, 0,1, then we have
|x 1 x 2 x 3 | 1.
It means that the ball B0, 0, 0,1 is an octahedron. Hence, Ba; r is an octahedron.
Remark: The exercise tells us one thing that Ba; r may not be an n ball if we
consider some different matrices.
3.28 Let d 1 and d 2 be the metrics of Exercise 3.27 and let x y denote the usual
Euclidean metric. Prove that the following inequalities for all x and y in R n :
d 1 x, y x y d 2 x, y and d 2 x, y n x y nd 1 x, y.
Proof: List the definitions of the three metrics, and compare with them as follows.