(New page: Metric Space (X,d) <math>d:X \times X \rightarrow \Re ^{+}</math> X is set, not necessarily vector space <math>x, y, z \in X</math> 1. <math>d(x,y)=d(y,x)</math> 2. <math>d(x,z)\leq d(...) |
|||
| Line 10: | Line 10: | ||
2. <math>d(x,z)\leq d(x,y)+d(y,z)</math> | 2. <math>d(x,z)\leq d(x,y)+d(y,z)</math> | ||
| − | 3. <math>d(x,y \geq 0, d(x,y)=0 \Leftrightarrow x=y)</math> | + | 3. <math>d(x,y) \geq 0, d(x,y)=0 \Leftrightarrow x=y)</math> |
| + | |||
| + | If X is vector space, metric can be induced by the norm <math>||\cdot||</math>. | ||
| + | |||
| + | <math>d(x,y)=||y-x||</math> | ||
| + | |||
| + | Norm is defined as follows | ||
| + | |||
| + | <math>||\cdot||: X \rightarrow \Re ^{+}</math> | ||
| + | |||
| + | 1. <math>|x| \geq 0, ||x||=0 \Leftrightarrow x=0</math> | ||
| + | 2. <math>||\alpha x||=|\alpha | ||x||</math> | ||
| + | 3. <math>||x+y|| \leq ||x|| + || ||</math> | ||
| + | |||
| + | Defining metric, we can measure similarity of elements of set X. | ||
| + | |||
| + | Example of metric | ||
| + | 1. Minkowski Metric ||<math> = \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p}</math> | ||
| + | 2. Riemannian Metric || | ||
| + | 3. | ||
Revision as of 11:54, 7 April 2008
Metric Space (X,d) $ d:X \times X \rightarrow \Re ^{+} $
X is set, not necessarily vector space
$ x, y, z \in X $
1. $ d(x,y)=d(y,x) $
2. $ d(x,z)\leq d(x,y)+d(y,z) $
3. $ d(x,y) \geq 0, d(x,y)=0 \Leftrightarrow x=y) $
If X is vector space, metric can be induced by the norm $ ||\cdot|| $.
$ d(x,y)=||y-x|| $
Norm is defined as follows
$ ||\cdot||: X \rightarrow \Re ^{+} $
1. $ |x| \geq 0, ||x||=0 \Leftrightarrow x=0 $ 2. $ ||\alpha x||=|\alpha | ||x|| $ 3. $ ||x+y|| \leq ||x|| + || || $
Defining metric, we can measure similarity of elements of set X.
Example of metric 1. Minkowski Metric ||$ = \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p} $ 2. Riemannian Metric || 3.
