Difference between revisions of "Complete metric space"

Latest revision as of 15:36, 24 November 2015

Given a metric space $(X,d)$ , if every Cauchy sequence converges to a limit (sequence) within $X$ then $X$ is a complete metric space^[1]^[2]. That is to say:

Given a sequence $(x_n)_{n=1}^\infty$ , it converging to a limit $x\in X$ or being a Cauchy sequence are equivalent. Or in symbols:
$\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n>N\implies d(x_n,x)]\iff\forall\epsilon>0\exists N\in\mathbb{N}\forall n,m\in\mathbb{N}[n\ge m>N\implies d(x_n,x_m)<\epsilon]$

Obviously the $\mathbb{R}$ (reals) are complete, considered with the usual topology induced by the Absolute value metric

A good example is the space of fractions, $\mathbb{Q}$ considered with the Absolute value metric again, there are rational sequences which converge to say, $\sqrt{2}$ , and $\sqrt{2}\notin\mathbb{Q}$
A better example is the space of continuous functions on an interval, C[a,b] and the distance function:
- $d(f,g)=\sqrt{\int^b_a\vert f(x)-g(x)\vert dx}$ for $f,g\in\mathcal{C}[a,b]$
- Let $a=-1$ and $b=1$ (WLOG)
- We can then see that the sequence of functions (fn)∞n=1 where each fn:[−1,1]→[0,1]⊂R given by:
  $f_n(x)=\left\{\begin{array}{lr}0 & \text{for }x\in[-1,0] \\ nx &\text{for }x\in(0,\frac{1}{n}] \\ 1 & \text{otherwise}\end{array}\right.$
  - Has a limit (note that: $\lim_{n\rightarrow\infty}(f_n)=f$ with $f(x)=\left\{\begin{array}{lr} 0 & \text{for }x\in[-1,0] \\ 1 & \text{otherwise}\end{array}\right.$ and that this $f$ isn't continuous (in $(\mathbb{R},\vert\cdot\vert)$ ) anyway!)
  - and that limit, $f$ isn't continuous, this we have shown that $\mathcal{C}[-1,1]$ isn't complete. (and by translation/scaling as needed, $\mathcal{C}[a,b]$ isn't complete)

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 ==Definition==
-Given a [[Metric space|metric space]] {{M|(X,d)}}, if every [[Cauchy sequence]] converges to a limit within {{M|X}} then {{M|X}} is a ''complete metric space''<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref>
+Given a [[Metric space|metric space]] {{M|(X,d)}}, if every [[Cauchy sequence]] converges to a [[limit (sequence)]] within {{M|X}} then {{M|X}} is a ''complete metric space''<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref><ref name="KMAPI">Analysis - Part I: Elements - Krzysztof Maurin</ref>. That is to say:
+* Given a [[sequence]] {{M|1=(x_n)_{n=1}^\infty}}, it converging to a limit {{M|x\in X}} or being a [[Cauchy sequence]] are equivalent. Or in symbols:
+* {{M|1=\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n>N\implies d(x_n,x)]\iff\forall\epsilon>0\exists N\in\mathbb{N}\forall n,m\in\mathbb{N}[n\ge m>N\implies d(x_n,x_m)<\epsilon]}}
 ==Examples==