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.
