Determine the functions d

Question

Metric Spaces of a nonempty set

Response to the following :

1. Show that the functions d defined below satisfy the properties of a metric.

a. Let X be any nonempty set and let d be defined

d(x,y)=   0 if x=y  and 1 if x≠y     ,  The d is the call the discrete metric.

b. If X is the set of all m-tuples of real numbers and, if for x=(α₁...α_m)  and y=(β₁...β_m),d(x,y)=max { |α_λ-β_λ|:k=1,...,m }  then (X,d) is a metric space.

c. Let X be the set of all real-valued frunctions which are defined and continuous on the closed interval [a,b] in and let

d(ƒ,g) =max { |ƒ^(t)-g^(t)|:tE [a,b]}