Replacing D by an ε-neighbourhood of f(D), it can be assumed that D is itself bounded in norm.
For z in D and v in X, set
where the supremum is taken over all holomorphic functions g on D with |g(z)| < 1.
Define the α-length of a piecewise differentiable curve γ:[0,1] D by
The Carathéodory metric is defined by
for x and y in D. It is a continuous function on D x D for the norm topology.
If the diameter of D is less than R then, by taking suitable holomorphic functions g of the form
with a in X* and b in C, it follows that
and hence that
In particular d defines a metric on D.
The chain rule
implies that
and hence f satisfies the following generalization of the Schwarz-Pick inequality:
For δ sufficiently small and y fixed in D, the same inequality can be applied to the holomorphic mapping
and yields the improved estimate:
The Banach fixed-point theorem can be applied to the restriction of f to the closure of f(D) on which d defines a complete metric, defining the same
topology as the norm.