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.
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This page is about the norm on a vector space. For other uses, see Definition:Norm.
Contents
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: | \(\displaystyle \forall x, y \in V:\) | \(\displaystyle \norm {x + y} \) | \(\displaystyle \le \) | \(\displaystyle \norm x + \norm y \) |
Normed Vector Space
Let $\norm {\,\cdot\,}$ be a norm on $V$.
Then $\struct {V, \norm {\,\cdot\,} }$ is a normed vector space.
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:
| \((N1)\) | $:$ | 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 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/Bounded Linear Transformation
- Definition:Norm/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: Chapter $1$: Normed and Banach spaces
