  This paper concerns the \textbf{abstract geometry of numbers}: namely the
pursuit of certain aspects of geometry of numbers over a suitable class of
normed domains. (The standard geometry of numbers is then viewed as geometry of
numbers over Z endowed with its standard absolute value.) In this work we study
normed domains of "linear type", in which an analogue of Minkowski's linear
forms theorem holds. We show that S-integer rings in number fields and
coordinate rings of (nice) affine algebraic curves over an arbitrary ground
field are of linear type. The theory is applied to quadratic forms in two ways,
yielding a Nullstellensatz and a Small Multiple Theorem.
