English Dictionary

Definition 2.3.1 A p-adic norm on vector space V over K is a function 𝛼:V→ R satisfying:

a) 𝛼(x)>0 and 𝛼(x)=0 if and only if x=0.

b) 𝛼(ax)=|a|𝛼(x) for a∈K and x∈V.

Definition 2.3.1 A p-adic norm on vector space V over K is a function 𝛼:V→ R satisfying: a) 𝛼(x)>0 and 𝛼(x)=0 if and only if x=0. b) 𝛼(ax)=|a|𝛼(x) for a∈K and x∈V. c) 𝛼(x+y)0, then the dilation t𝛼 is a p-adic norm, and we denote by N_p(V) the set of dilation classes of p-adic norms on V.