Category:Definitions/Examples of Metric Spaces
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This category contains definitions of examples of Metric Space.
A metric space $M = \struct {A, d}$ is an ordered pair consisting of:
together with:
- $(2): \quad$ a real-valued function $d: A \times A \to \R$ which acts on $A$, satisfying the metric space axioms:
\((\text M 1)\) | $:$ | \(\ds \forall x \in A:\) | \(\ds \map d {x, x} = 0 \) | ||||||
\((\text M 2)\) | $:$ | Triangle Inequality: | \(\ds \forall x, y, z \in A:\) | \(\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z} \) | |||||
\((\text M 3)\) | $:$ | \(\ds \forall x, y \in A:\) | \(\ds \map d {x, y} = \map d {y, x} \) | ||||||
\((\text M 4)\) | $:$ | \(\ds \forall x, y \in A:\) | \(\ds x \ne y \implies \map d {x, y} > 0 \) |
Subcategories
This category has the following 14 subcategories, out of 14 total.
C
E
- Definitions/Euclidean Metric (17 P)
H
- Definitions/Hilbert Cube (3 P)
L
- Definitions/L1 Metric (3 P)
- Definitions/L2 Metric (3 P)
P
- Definitions/P-adic Metrics (4 P)
R
S
- Definitions/Supremum Metric (12 P)
T
- Definitions/Taxicab Metric (8 P)
Pages in category "Definitions/Examples of Metric Spaces"
The following 27 pages are in this category, out of 27 total.