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[ilmath]d:X\times X\rightarrow\mathbb{R}_{\ge 0} [/ilmath]
Where [ilmath]X[/ilmath] is any set
relation to other topological spaces
is a
contains all
Related objects
Induced by norm
  • [ilmath]d_{\Vert\cdot\Vert}:V\times V\rightarrow\mathbb{R}_{\ge 0}[/ilmath]
  • [ilmath]d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert[/ilmath]

For [ilmath]V[/ilmath] a vector space over [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath]

Induced by inner product

An inner product induces a norm:

  • [ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}[/ilmath]
  • [ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}[/ilmath]

Which induces a metric:

  • [ilmath]d_{\langle\cdot,\cdot\rangle}:V\times V\rightarrow\mathbb{R}_{\ge 0}[/ilmath]
  • [ilmath]d_{\langle\cdot,\cdot\rangle}:(x,y)\mapsto\sqrt{\langle x-y,x-y\rangle}[/ilmath]
A metric is the most abstract notion of distance. It requires no structure on the underlying set.



TODO: Separate metric and metric space