Definition:Norm (Division Ring)
This page is about the norm on a division ring or a field. For other uses, see Definition:Norm.
Definition
Let $\left({R, +, \circ}\right)$ be a division ring, and denote its zero by $0_R$.
A norm on $R$ is a map on $R$ to the nonnegative reals $\left\Vert{\cdot}\right\Vert: R \to \R_+ \cup \left\{{0}\right\}$ satisfying the following three properties (for all $x,y \in R$):
| N1: | Positive definiteness: | $\left\Vert {x} \right\Vert = 0 \iff x = 0_R$ |
| N2: | Multiplicativity: | $\left\Vert{x y}\right\Vert = \left\Vert{x}\right\Vert \times \left\Vert{y}\right\Vert$ |
| N3: | Triangle inequality: | $\left\Vert {x + y}\right\Vert \leq \left\Vert{x}\right\Vert + \left\Vert{y}\right\Vert$ |
These may be referred to as the (division ring) norm axioms.
Some authors refer to this concept as an (abstract) absolute value on $R$.
Also, a field $k$ that is endowed with a norm is known as a valued field.
Notes
In the literature, it is more common to define the norm only for subfields of the complex numbers.
However, the definition given here incorporates this approach.
some links to literature be given
You can help ProofWiki by adding this information.
To discuss it in more detail, feel free to use the talk page.
If you are able to do this, then when you have done so you can remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Also See
- Definition:Modulus of Complex Number, a well-known example of a norm.
- Definition:Norm (Vector Space), an extension to vector spaces.
