Let X be the metric space, and p∈X. Define a map T: BC(X) → R by

T(f) = f(p)for every f∈BC(X). Show that T is Lipschitz, and Lip(T) ≤ 1 (with the sup norm on BC(X) and the usual metric on R).

Here, I have to show that

∥T(f) - T(g)∥∞ ≤ λ│f(p) – g(p)│

is that right ?

but how do I define ∥T(f) - T(g)∥∞ ?

and what does Lip(T) mean ?