Metric Spaces and Banach Fixed Point Theorem

Definition 1.2.1 Let X be a non-empty set and d a function defined on X×X into the set of real numbers R

satisfying the following conditions:

(i) d(x,y)=0 if and only if x=y, (ii) d(x,y)=d(y,x) for all x, and (iii) d(x, y)≤d(x,z)+d(z,y) for all x,y, .