Category:Definitions/Examples of Norms
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This category contains definitions of examples of norms.
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: | \(\ds \forall x, y \in V:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |
Subcategories
This category has the following 6 subcategories, out of 6 total.
C
- Definitions/Complex Modulus (2 P)
P
- Definitions/P-Adic Norms (empty)
- Definitions/P-adic Norms (4 P)
- Definitions/P-Norms (5 P)
T
- Definitions/Taxicab Norm (2 P)
- Definitions/Trivial Norms (5 P)
Pages in category "Definitions/Examples of Norms"
The following 11 pages are in this category, out of 11 total.