Difference between revisions of "The real numbers"

Revision as of 21:29, 26 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
• 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 $\mathbb{R}^n$ and $\mathbb{R}$

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

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

References

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