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 $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$.
A (multiplicative) norm on $R$ is a mapping from $R$ to the non-negative reals:
- $\norm {\,\cdot\,}: R \to \R_{\ge 0}$
satisfying the (ring) multiplicative norm axioms:
| \((N1)\) | $:$ | Positive Definiteness: | \(\displaystyle \forall x \in R:\) | \(\displaystyle \norm x = 0 \) | \(\displaystyle \iff \) | \(\displaystyle x = 0_R \) | ||
| \((N2)\) | $:$ | Multiplicativity: | \(\displaystyle \forall x, y \in R:\) | \(\displaystyle \norm {x \circ y} \) | \(\displaystyle = \) | \(\displaystyle \norm x \times \norm y \) | ||
| \((N3)\) | $:$ | Triangle Inequality: | \(\displaystyle \forall x, y \in R:\) | \(\displaystyle \norm {x + y} \) | \(\displaystyle \le \) | \(\displaystyle \norm x + \norm y \) |
Notes
In contrast to the definition of a norm on a division ring, a ring norm is always assumed to be a submultiplicative norm.
The reason for this is by Normed Vector Space Requires Multiplicative Norm on Division Ring, the norm on a division ring that is the scalar ring of a normed vector space must be a multiplicative norm.
By Ring with Multiplicative Norm has No Proper Zero Divisors it follows that a ring with zero divisors has no multiplicative norms, so a multiplicative norm is too restrictive for a general ring.
Also known as
Some authors refer to this concept as an (abstract) absolute value on $R$.
A field that is endowed with a norm is thereby referred as a valued field.
Also defined as
In the literature, it is 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
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Also see
- Definition:Absolute Value, a well known norm as shown in Absolute Value is Norm.
- Definition:Complex Modulus, a well known norm as shown in Complex Modulus is Norm.
- Definition:Field Norm of Quaternion, which is actually not a norm as shown in Field Norm of Quaternion is not Norm.
- Definition:Norm on Vector Space
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (next): $\S 2.1$: Absolute Values on a Field: Definition $2.1.1$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (next): $\S 1.2$: Normed Fields: Definition $1.5$
