Difference between revisions of "The real numbers"

Revision as of 17:38, 17 February 2017

The real line is the name given to the reals with their "usual topology", the topology that is induced by the absolute value metric

Definition

Cantor's construction of the real numbers

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

• $\mathscr{C}$ - the set of all Cauchy sequences in $\mathbb{Q}$ - the quotients
• $\sim$ - 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 $x\in\mathcal{Q}$ with the sequence $({ x_n })_{ n = 1 }^{ \infty }\subseteq \mathbb{Q}$ where $\forall n\in\mathbb{N}[x_n:=x]$ we can embed $\mathbb{Q}$ in $\mathbb{R}:=\mathscr{C}/\sim$

Axiomatic construction of the real numbers

Axiomatic construction of the real numbers/Definition

$\mathbb{R}$ is an example of:

• Vector space
• Field ($\implies\ \ldots\implies$ ring)
• Complete metric space ($\implies$ topological space)
  • With the metric of absolute value

TODO: Flesh out

Properties

Notes

References

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