Normed field

Summary

A normed field is a field with an absolute value, see the definition. An absolute value is a size function that is zero only at $0$, is multiplicative and satisfies the triangle inequality. The size gives a distance, so a normed field is a metric space and a topological field. The real and complex numbers with the usual absolute value and the rationals with a p-adic absolute value are the basic examples. Normed fields are the scalars of normed vector spaces in general functional analysis.