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We introduce a function model for the Teichm\"uller space of a closedhyperbolic Riemann surface. Then we introduce a new metric by using the maximumnorm on the function space on the Teichm\"uller space. We prove that theidentity map from the Teichm\"uller space equipped with the usual Teichm\"ullermetric to the Teichm\"uller space equipped with this new metric is uniformlycontinuous. Furthermore, we also prove that the inverse of the identity, thatis, the identity map from the Teichm\"uller space equipped with this new metricto the Teichm\"uller space equipped with the usual Teichm\"uller metric, iscontinuous. Therefore, the topology induced by the new metric is just the sameas the topology induced by the usual Teichm\"uller metric on the Teichm\"ullerspace. We give a remark about the pressure metric and the Weil-Peterssonmetric.

Function Model of the Teichm\"uller space of a closed hyperbolic Riemann Surface