Definition:Norm/Vector Space
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: | \(\ds \forall x \in V:\) | \(\ds \norm x = 0 \) | \(\ds \iff \) | \(\ds x = \mathbf 0_V \) | |||
| \((\text N 2)\) | $:$ | Positive Homogeneity: | \(\ds \forall x \in V, \lambda \in R:\) | \(\ds \norm {\lambda x} \) | \(\ds = \) | \(\ds \norm {\lambda}_R \times \norm x \) | |||
| \((\text N 3)\) | $:$ | Triangle Inequality: | \(\ds \forall x, y \in V:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |
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: | \(\ds \forall x \in R:\) | \(\ds \norm x = 0 \) | \(\ds \iff \) | \(\ds x = 0_R \) | |||
| \((\text N 2)\) | $:$ | Multiplicativity: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x \circ y} \) | \(\ds = \) | \(\ds \norm x \times \norm y \) | |||
| \((\text N 3)\) | $:$ | Triangle Inequality: | \(\ds \forall x, y \in R:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |
Also presented as
The norm axioms for a vector space are usually defined over the real numbers $\R$ or complex numbers $\C$, and so presented in the form:
| \((\text N 1)\) | $:$ | Positive Definiteness: | \(\ds \forall x \in V:\) | \(\ds \norm x = 0 \) | \(\ds \iff \) | \(\ds x = \mathbf 0_V \) | |||
| \((\text N 2)\) | $:$ | Positive Homogeneity: | \(\ds \forall x \in V, \lambda \in \GF:\) | \(\ds \norm {\lambda x} \) | \(\ds = \) | \(\ds \size \lambda \norm x \) | where $\GF \in \set {\R, \C}$ | ||
| \((\text N 3)\) | $:$ | Triangle Inequality: | \(\ds \forall x, y \in V:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |
where $\size \lambda$ denotes either:
- the absolute value function on $\R$
- the complex modulus on $\C$.
The equivalence of the two definitions is confirmed by:
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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): norm: 1. (of a vector space)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): norm: 1. (of a vector space)
- 2011: Graham R. Allan and H. Garth Dales: Introduction to Banach Spaces and Algebras ... (previous) ... (next): $2.1$: Normed Spaces
- 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
- 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $3.1$: Norms