Difference between revisions of "The real numbers"

	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;

Revision as of 21:29, 26 February 2017

Stub grade: C

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Once cleaned up and fleshed out, demote to D

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]

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

[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:

that the familiar operations of addition, multiplication and division are well defined and
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]

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

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]:

@@ Line 1: / Line 1: @@
 {{Stub page|grade=C|msg=Once cleaned up and fleshed out, demote to D
 * Be sure to include [[Example:The real line with the finite complement topology is not Hausdorff]]}}
-<div style="float:right;margin:0px;margin-left:0.2em;">{{Infobox|style=max-width:30ex;|title=The real numbers|above=<div style="max-width:25em;"><span style="font-size:9em;">{{M|\mathbb{R} }}</span></div>}}</div>
+<div style="float:right;margin:0px;margin-left:0.2em;">{{/Infobox}}</div>
-: [[The real line]] is the name given to the reals with their "usual topology", the [[topology]] that is [[topology induced by a metric|induced]] by the [[absolute value metric]]
+:* [[The real line]] is the name given to the reals with their "usual topology", the [[topology]] that is [[topology induced by a metric|induced]] by the [[absolute value metric]]
+:* [[Borel sigma-algebra of the real line]] - useful in [[Measure Theory (subject)|Measure Theory]] although distinct from [[Lebesgue measurable sets]] on the real line
+:** {{XXX|Pages needed}} for the Lebesgue-measurable structure on {{M|\mathbb{R}^n}} and {{M|\mathbb{R} }}
 __TOC__
 ==Definition==

Algebraic structure
The real numbers
[ilmath]\mathbb{R} [/ilmath]
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