distance

[[concept]]

A metric or distance is a function $d:M\times M\to \mathbb{R}$ that satisfies

1. $d(x,x)=0$
2. $d(x,y)>0\forall x\ne y$
3. $d(x,y)=d(y,x)$
4. $d(x,z)\le d(x,y)+d(y,z)$

This is part of the definition of a metric space.

functional analysis

Metric

If $X$ is a set then $d:X\times X\to [0,\infty )$ is a metric if

1. (identifiability) $d(x,y)=0⟺x=y$
2. (symmetry) $\forall x,y\in X,d(x,y)=d(y,x)$
3. (triangle inequality) $\forall x,y,z\in X,d(x,y)\le d(x,z)+d(z,y)$

Reference

see also

• norm
• metric space

Mentions

Mentions

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