The real numbers

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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


Cantor's construction of the real numbers

The set of real numbers, [ilmath]\mathbb{R} [/ilmath], is the quotient space, [ilmath]\mathscr{C}/\sim[/ilmath] where:[1]

We further claim:

  1. that the familiar operations of addition, multiplication and division are well defined and
  2. by associating [ilmath]x\in\mathcal{Q} [/ilmath] with the sequence [ilmath] ({ x_n })_{ n = 1 }^{ \infty }\subseteq \mathbb{Q} [/ilmath] where [ilmath]\forall n\in\mathbb{N}[x_n:=x][/ilmath] we can embed [ilmath]\mathbb{Q} [/ilmath] in [ilmath]\mathbb{R}:=\mathscr{C}/\sim[/ilmath]

Axiomatic construction of the real numbers

Axiomatic construction of the real numbers/Definition

[ilmath]\mathbb{R} [/ilmath] is an example of:

TODO: Flesh out


If [ilmath]S\subseteq\mathbb{R} [/ilmath] is a non-empty set of real numbers that has an upper bound then[2]:

  • [ilmath]\text{Sup}(S)[/ilmath] (the supremum of [ilmath]S[/ilmath]) exists.


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


  1. Analysis - Part 1: Elements - Krzysztof Maurin
  2. Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha