1. Let X = × × = 3 and define the metric d: 3 × 3 → byd ((x1, y1, z1), (x2 , y2 , z2 )) = | x1 − x2 | + | y1 − y2 | + | z1 − z2 | .Describe the neighborhood N (0 ; 1), where 0 is the origin in 32. Let F be a nonempty set of functions that map [0, 1] into [0, 1]. For f and g in F, defined( f , g) = sup {| f (x) − g(x) | : x∈[0,1]}.Show that d is a metric on F.

1. Let X = × × = 3 and define the metric d: 3 × 3 → by d ((x1, y1, z1), (x2 , y2 , z2 )) = | x1 − x2 | + | y1 − y2 | + | z1 − z2 | . Describe the neighborhood N (0 ; 1), where 0 is the origin in 3...

1. Let X =

×

×

=

³
and define the metric d:

³
×

³
→

by

d ((x₁, y₁, z₁), (x₂
, y₂
, z₂
)) = | x₁
− x₂
| + | y₁
− y₂
| + | z₁
− z₂
| .

Describe the neighborhood N (0 ; 1), where 0 is the origin in

³

2. Let F be a nonempty set of functions that map [0, 1] into [0, 1]. For f and g in F, define

d( f , g) = sup {| f (x) − g(x) | : x∈[0,1]}.

Show that d is a metric on F.

May 05, 2022