Axiom:Vector Space Norm Axioms
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This page is about the norm axioms on a vector space. For other uses, see Norm Axioms.
Definition
Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$.
Let $V$ be a vector space over $R$, with zero $\mathbf 0_V$.
Let $\norm {\,\cdot\,}: V \to \R_{\ge 0}$ be a norm on $V$.
The norm axioms are the following conditions on $\norm {\,\cdot\,}$ which define $\norm {\,\cdot\,}$ as being a norm:
\((\text N 1)\) | $:$ | 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: | \(\ds \forall x, y \in V:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |
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 see
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)