The real numbers/Infobox

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The real numbers
[ilmath]\mathbb{R} [/ilmath]
Algebraic structure
TODO: Todo
- is a field
Standard topological structures
Main page: The real line
inner product [ilmath]\langle a,b\rangle:\eq a*b[/ilmath]
- Euclidean inner-product on [ilmath]\mathbb{R}^1[/ilmath]
norm [ilmath]\Vert x\Vert:\eq\sqrt{\langle x,x\rangle}\eq\vert x\vert[/ilmath]
- Euclidean norm on [ilmath]\mathbb{R}^1[/ilmath]
metric [ilmath]d(x,y):\eq\Vert x-y\Vert\eq \vert x-y\vert[/ilmath]
- Absolute value
- Euclidean metric on [ilmath]\mathbb{R}^1[/ilmath]
topology topology induced by the metric [ilmath]d[/ilmath]
Standard measure-theoretic structures
measurable space Borel [ilmath]\sigma[/ilmath]-algebra of [ilmath]\mathbb{R} [/ilmath][Note 1]
- other:
Lebesgue-measurable sets of [ilmath]\mathbb{R} [/ilmath]
  • contains the Borel [ilmath]\sigma[/ilmath]-algebra

The real line discusses [ilmath]\mathbb{R} [/ilmath] as a set.


  1. This is just the Borel sigma-algebra on the real line (with its usual topology)