f$_{a}$(x)=d(a,x)−d(a$_{0}$,x) is a bounded continuous map, i.e. f$_{a}$∈F$_{b}$(X;R)∩C(X;R)

Question

Let (X,d) be a metric space and fix a point a$_{0}$∈X.

a)Prove that for every a∈X, the map f$_{a}$:(X,d)→(R,|⋅|), given by f$_{a}$(x)=d(a,x)−d(a$_{0}$,x) is a bounded continuous map, i.e. f$_{a}$∈F$_{b}$(X;R)∩C(X;R).

b)Prove that the map Φ:(X,d)→(F$_{b}$(X,R),ρ) given by Φ(a)=f$_{a}$ is an isometry

I'm honestly not even sure where to start with this question, any help with even starting would be appreciated!

Answer 1

(a) Hint: show, by definition, that the map that sends x to d(a,x) is continuous. From this you can deduce your function is continuous, and hence bounded.

(b) To answer this I need to know what is the space (Fb,b(X,R),ρ)? More specifically, what is the metric \rho?