---
tag: ZFW9Q3WTAQ
kind: concept
name: Normed field
msc: [12J05, 12J10]
specializes: [D7YQW9RG24]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_value_(algebra)
---

## Summary

A normed field is a [[field|04C5ZBG1WT]] with an absolute value, see the [[definition|65HSHAH2Y4]].
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|0VKD1TP7F3]] and a [[topological field|D7YQW9RG24]].
The real and complex numbers with the usual [[absolute value|0FFXKTKKQC]] and the rationals with a [[p-adic absolute value|WGN82TCEF2]] are the basic examples.
Normed fields are the scalars of [[normed vector spaces|03J5RAKB30]] in general functional analysis.

## Explanations

### L1 human

The absolute value of real numbers tells how big a number is, and $\abs{x - y}$ tells how far apart two numbers are.
A normed field is any field with such a size function that behaves well with products and sums.
The size of a product is the product of the sizes, and the size of a sum is at most the sum of the sizes.
For example, on $\Q$ one may call a number small when it has many factors of $3$: $\abs{9}_3 = 1/9$ and $\abs{1/3}_3 = 3$.