  We introduce a function model for the Teichm\"uller space of a closed
hyperbolic Riemann surface. Then we introduce a new metric by using the maximum
norm on the function space on the Teichm\"uller space. We prove that the
identity map from the Teichm\"uller space equipped with the usual Teichm\"uller
metric to the Teichm\"uller space equipped with this new metric is uniformly
continuous. Furthermore, we also prove that the inverse of the identity, that
is, the identity map from the Teichm\"uller space equipped with this new metric
to the Teichm\"uller space equipped with the usual Teichm\"uller metric, is
continuous. Therefore, the topology induced by the new metric is just the same
as the topology induced by the usual Teichm\"uller metric on the Teichm\"uller
space. We give a remark about the pressure metric and the Weil-Petersson
metric.
