(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(...)
 
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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>
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3. <math>d(x,y) \geq 0, d(x,y)=0 \Leftrightarrow x=y)</math>
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If X is vector space, metric can be induced by the norm <math>||\cdot||</math>.
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<math>d(x,y)=||y-x||</math>
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Norm is defined as follows
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<math>||\cdot||: X \rightarrow \Re ^{+}</math>
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1. <math>|x| \geq 0, ||x||=0 \Leftrightarrow x=0</math>
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2. <math>||\alpha x||=|\alpha | ||x||</math>
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3. <math>||x+y|| \leq ||x|| + || ||</math>
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Defining metric, we can measure similarity of elements of set X.
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Example of metric
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1. Minkowski Metric ||<math> = \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p}</math>
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2. Riemannian Metric ||
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3.

Revision as of 12:54, 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.

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