Definition:Norm/Vector Space
Add an alternative (simplified) definition for vector spaces over usual number fields based on the Notes section. Prominently link to something about standard norms on real (and complex?) finite-dimensional vector spaces.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ 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).
This page is about Norm on Vector Space. For other uses, see Norm.
Definition
Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$.
Let $V$ be a vector space over $R$, with zero $0_V$.
A norm on $V$ is a map from $V$ to the nonnegative reals:
- $\norm{\,\cdot\,}: V \to \R_{\ge 0}$
satisfying the (vector space) norm axioms:
| \((\text N 1)\) | $:$ | Positive definiteness: | \(\displaystyle \forall x \in V:\) | \(\displaystyle \norm x = 0 \) | \(\displaystyle \iff \) | \(\displaystyle x = \mathbf 0_V \) | ||
| \((\text N 2)\) | $:$ | Positive homogeneity: | \(\displaystyle \forall x \in V, \lambda \in R:\) | \(\displaystyle \norm {\lambda x} \) | \(\displaystyle = \) | \(\displaystyle \norm {\lambda}_R \times \norm x \) | ||
| \((\text N 3)\) | $:$ | Triangle inequality: | \(\displaystyle \forall x, y \in V:\) | \(\displaystyle \norm {x + y} \) | \(\displaystyle \le \) | \(\displaystyle \norm x + \norm y \) |
Division Ring
When the vector space $V$ is the $R$-vector space $R$, the definition reduces to the division ring norm:
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:
| \((\text N 1)\) | $:$ | Positive Definiteness: | \(\displaystyle \forall x \in R:\) | \(\displaystyle \norm x = 0 \) | \(\displaystyle \iff \) | \(\displaystyle x = 0_R \) | ||
| \((\text N 2)\) | $:$ | Multiplicativity: | \(\displaystyle \forall x, y \in R:\) | \(\displaystyle \norm {x \circ y} \) | \(\displaystyle = \) | \(\displaystyle \norm x \times \norm y \) | ||
| \((\text N 3)\) | $:$ | Triangle Inequality: | \(\displaystyle \forall x, y \in R:\) | \(\displaystyle \norm {x + y} \) | \(\displaystyle \le \) | \(\displaystyle \norm x + \norm y \) |
Notes
Replace this with something precise that proves the definitions are consistent.
Until this has been finished, please leave {{refactor}} in the code.
New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.
In the literature, it is more common to define the norm only if $R$ is $\R$ or $\C$ (and consequently $\norm {\,\cdot\,}_R$ is the absolute value or modulus function respectively).
However, the definition given here incorporates this approach.
Also known as
The term length is occasionally seen as an alternative for norm.
Also see
- Definition:Norm on Division Ring
- Definition:Norm on Algebra
- Definition:Norm on Bounded Linear Transformation
- Definition:Norm on Bounded Linear Functional
Sources
- 2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control ... (next): $1.1$: Basic Definitions
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed spaces
