[tex]\|f_x-f_y\|[/tex] gives you .
It seems that the claim is obvious once you know that . Do you understand why this fact holds?
Consider a metric space (X, d). Let (B(X), ||.||) be the vector space of bounded real valued functions on X with the sup norm. Denote fx(y) = d(x, y). Fix point a in X and let gx=fx – fa. Show that x-> gx is an isometric embedding of X into B(X). Note that d(x, y) = ||fx – fy||
I uploaded a picture of the problem since I was unable to introduce the subscripts properly (sorry still new in the forum).
If someone could give me an idea on how to solve it, I would be really greatful.
Thanks in advance.