Normed Graded Linear Series over a Trivially Valued Field

This chapter deals with normed graded linear series over a trivially valued field. We first remind in Sect. 3.1 the geometry of adelic vector bundles on the adelic curve consisting of a single copy of the trivial absolute value on a field. Then, we introduce in Sect. 3.2 the notion of normed graded algebra over such adelic curves and discuss the properties of the associated spectrum norm. The last two sections are devoted to the study of asymptotic behaviours of normed graded linear series.