Bounded sections on a Riemann surface

Abstract:

Let X denote a hyperbolic Riemann surface, $\zeta$ a unitary line bundle, and ${H^\infty }(\zeta )$ the Banach space of bounded holomorphic sections of $\zeta$. If, for a given point $\xi$ in X, the norm of the evaluation functional on ${H^\infty }(\zeta )$ varies continuously with the bundle $\zeta$, then it is shown that the space of bounded holomorphic sections is dense in the space of holomorphic sections for every unitary line bundle.