Metric Space

analysis

Definition

Metric Space

A set $M$ with a mapping $d:M\times M\to \mathbb{R}$ is called metric space $⟨M,d⟩$ iff the mapping $d$, called the metric of $M$, has the following properties:

1. $\forall x,y\in M:d(x,y)\ge 0$
2. $\forall x,y\in M:d(x,y)=0⟺x=y$
3. $\forall x,y\in M:d(x,y)=d(y,x)$
4. $\forall x,y,z\in M:d(x,y)\le d(x,z)+d(z,y)$

Metrics

• $d_{\alpha }(\mathbf{x},\mathbf{y})={\left({\sum }_{i=1}^n|x_i-y_i{|}^{\alpha }\right)}^{1/\alpha }$ with $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^n$ and $1\le \alpha \le \infty$
• Sum: $d_1(\mathbf{x},\mathbf{y})={\sum }_{i=1}^n|x_i-y_i|$ with $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^n$
• Euclidean: $d_2(\mathbf{x},\mathbf{y})=\sqrt{{\sum }_{i=1}^n(x_i-y_i{)}^2}$ with $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^n$
• Max: $d_{\infty }(\mathbf{x},\mathbf{y})={\max }_{1\le i\le n}|x_i-y_i|$ with $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^n$
• Hamming: $d_H(\mathbf{x},\mathbf{y})=|\{x_i\ne y_i{\}}_{1\le i\le n}|$

Neighbourhood

An neighbourhood, analogous to epsilon neighbourhood, is defined as:

$$K_d(x,r)=\{y\in X\mid d(x,y)<r\}$$