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.