Exercise 6.1.26

Answers

1.

d (x, y) = ∥ x - y ∥ \geq 0 .

2.

d (x, y) = ∥ x - y ∥ = ∥ y - x ∥ = d (y, x) .

3.

d (x, y) = ∥ x - y ∥ = ∥ (x - z) + (z - y) ∥ \leq ∥ x - z ∥ + ∥ z - y ∥ = d (x, y) + d (z, y) .

4.

d (x, x) = ∥ x - x ∥ = ∥ 0 ∥ = 0 .

5.

d (x, y) = ∥ x - y ∥ > 0

if $x - y$ is not zero. That is, the distance is not zero if $x \neq y$ .

jephianlin

2011-06-27 00:00