View source for Complete metric space

==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>

==Examples==
===Complete space===
* Obviously the {{M|\mathbb{R} }} (reals) are complete, considered with the usual topology induced by the [[Absolute value]] metric
===Incomplete space===
* A good example is the space of fractions, {{M|\mathbb{Q} }} considered with the [[Absolute value]] metric again, there are rational sequences which converge to say, {{M|\sqrt{2} }}, and {{M|\sqrt{2}\notin\mathbb{Q} }}
* A better example is the [[Space of continuous functions on an interval|space of continuous functions on an interval]], {{M|\mathcal{C}[a,b]}} and the distance function:
** {{M|1=d(f,g)=\sqrt{\int^b_a\vert f(x)-g(x)\vert dx} }} for {{M|f,g\in\mathcal{C}[a,b]}}
** Let {{M|1=a=-1}} and {{M|1=b=1}} (WLOG)
** We can then see that the sequence of functions {{M|1=(f_n)_{n=1}^\infty}} where each {{M|f_n:[-1,1]\rightarrow[0,1]\subset\mathbb{R} }} given by:
**: {{M|1=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: {{M|1=\lim_{n\rightarrow\infty}(f_n)=f}} with {{M|1=f(x)=\left\{\begin{array}{lr} 0 & \text{for }x\in[-1,0] \\ 1 & \text{otherwise}\end{array}\right.}} and that this {{M|f}} isn't continuous (in {{M|(\mathbb{R},\vert\cdot\vert)}}) anyway!)
*** and that limit, {{M|f}} isn't continuous, this we have shown that {{M|\mathcal{C}[-1,1]}} isn't complete. (and by translation/scaling as needed, {{M|\mathcal{C}[a,b]}} isn't complete)
==References==
<references/>

{{Definition|Topology|Metric Space}}