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]
- other:	Lebesgue-measurable sets of [ilmath]\mathbb{R} [/ilmath] contains the Borel [ilmath]\sigma[/ilmath]-algebra

• The real line is the name given to the reals with their "usual topology", the topology that is induced by the absolute value metric
• Borel sigma-algebra of the real line - useful in Measure Theory although distinct from Lebesgue measurable sets on the real line
  • TODO: Pages needed
    for the Lebesgue-measurable structure on [ilmath]\mathbb{R}^n[/ilmath] and [ilmath]\mathbb{R} [/ilmath]

Definition

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:

• [ilmath]\mathscr{C} [/ilmath] - the set of all Cauchy sequences in [ilmath]\mathbb{Q} [/ilmath] - the quotients
• [ilmath]\sim[/ilmath] - the usual equivalence of Cauchy sequences

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:

• Vector space
• Field ([ilmath]\implies\ \ldots\implies[/ilmath] ring)
• Complete metric space ([ilmath]\implies[/ilmath] topological space)
  • With the metric of absolute value

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TODO: Flesh out

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Properties

• The axiom of completeness - a badly named property that isn't really an axiom.

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

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