Category:Triangle Inequality
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This category contains results about the Triangle Inequality.
Triangle Inequality on Metric Space
Recall the metric space axioms:
| \((\text M 1)\) | $:$ | \(\ds \forall x \in A:\) | \(\ds \map d {x, x} = 0 \) | ||||||
| \((\text M 2)\) | $:$ | Triangle Inequality: | \(\ds \forall x, y, z \in A:\) | \(\ds \map d {x, y} + \map d {y, z} \ge \map d {x, z} \) | |||||
| \((\text M 3)\) | $:$ | \(\ds \forall x, y \in A:\) | \(\ds \map d {x, y} = \map d {y, x} \) | ||||||
| \((\text M 4)\) | $:$ | \(\ds \forall x, y \in A:\) | \(\ds x \ne y \implies \map d {x, y} > 0 \) |
Axiom $\text M 2$ is referred to as the triangle inequality, as it is a generalization of the Triangle Inequality which holds on the real number line and complex plane.
Triangle Inequality on Normed Vector Space
Recall 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: | \(\ds \forall x, y \in V:\) | \(\ds \norm {x + y} \) | \(\ds \le \) | \(\ds \norm x + \norm y \) |
Axiom $\text N 3$ is referred to as the triangle inequality, as it is a generalization of the Triangle Inequality which holds on the real number line and complex plane.
Triangle Inequality on Matrix Space
Recall the matrix norm axioms:
| \((\text N 1)\) | $:$ | Positive Definiteness: | \(\ds \forall \mathbf A \in \map {\MM_\GF} {m, n}:\) | \(\ds \norm {\mathbf A} = 0 \) | \(\ds \iff \) | \(\ds \mathbf A = \mathbf 0_{m, n} \) | where $\mathbf 0_{m, n}$ denotes the zero matrix of order $m \times n$ | ||
| \((\text N 2)\) | $:$ | Positive Homogeneity: | \(\ds \forall x \in \map {\MM_\GF} {m, n}, \lambda \in \GF:\) | \(\ds \norm {\lambda \mathbf A} \) | \(\ds = \) | \(\ds \norm \lambda \times \norm {\mathbf A} \) | where $\norm \lambda$ denotes the (division ring) norm of $\lambda$ | ||
| \((\text N 3)\) | $:$ | Triangle Inequality: | \(\ds \forall \mathbf A, \mathbf B \in \map {\MM_\GF} {m, n}:\) | \(\ds \norm {\mathbf A + \mathbf B} \) | \(\ds \le \) | \(\ds \norm {\mathbf A} + \norm {\mathbf B} \) |
Axiom $\text N 3$ is referred to as the triangle inequality, as it is a generalization of the Triangle Inequality which holds on the real number line and complex plane.
Subcategories
This category has the following 2 subcategories, out of 2 total.
E
T
Pages in category "Triangle Inequality"
The following 51 pages are in this category, out of 51 total.
R
- Reverse Triangle Inequality
- Reverse Triangle Inequality on Normed Division Ring
- Reverse Triangle Inequality on Real Numbers
- Reverse Triangle Inequality/Normed Division Ring
- Reverse Triangle Inequality/Normed Vector Space
- Reverse Triangle Inequality/Real and Complex Fields
- Reverse Triangle Inequality/Real and Complex Fields/Corollary 1
- Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 1
- Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 2
- Reverse Triangle Inequality/Real and Complex Fields/Corollary 1/Proof 3
- Reverse Triangle Inequality/Real and Complex Fields/Corollary 2
- Reverse Triangle Inequality/Real and Complex Fields/Corollary 3
- Reverse Triangle Inequality/Real and Complex Fields/Proof 1
- Reverse Triangle Inequality/Real and Complex Fields/Proof 2
- Reverse Triangle Inequality/Seminormed Vector Space
T
- Triangle Inequality
- Triangle Inequality for Complex Numbers
- Triangle Inequality for Complex Numbers/Corollary 3
- Triangle Inequality for Generalized Sums
- Triangle Inequality for Indexed Summations
- Triangle Inequality for Integrals
- Triangle Inequality for Real Numbers
- Triangle Inequality for Series
- Triangle Inequality for Series/Lebesgue Spaces
- Triangle Inequality for Summation over Finite Set
- Triangle Inequality for Variation of Complex Measure
- Triangle Inequality for Vectors in Euclidean Space
- Triangle Inequality on Distance from Point to Subset
- Triangle Inequality/Complex Numbers
- Triangle Inequality/Complex Numbers/Examples
- Triangle Inequality/Complex Numbers/Examples/3 Arguments
- Triangle Inequality/Complex Numbers/Examples/3 Arguments/Proof 1
- Triangle Inequality/Complex Numbers/Examples/3 Arguments/Proof 2
- Triangle Inequality/Complex Numbers/Examples/3 Arguments/Proof 3
- Triangle Inequality/Complex Numbers/General Result
- Triangle Inequality/Complex Numbers/Proof 1
- Triangle Inequality/Complex Numbers/Proof 2
- Triangle Inequality/Complex Numbers/Proof 3
- Triangle Inequality/Complex Numbers/Proof 4
- Triangle Inequality/Geometry
- Triangle Inequality/Real Numbers
- Triangle Inequality/Real Numbers/General Result
- Triangle Inequality/Real Numbers/Proof 1
- Triangle Inequality/Real Numbers/Proof 2
- Triangle Inequality/Real Numbers/Proof 3
- Triangle Inequality/Real Numbers/Proof 4
- Triangle Inequality/Real Numbers/Proof 5
- Triangle Inequality/Vectors in Euclidean Space