4
$\begingroup$

How can I show that the space of continuous real valued functions on R with compact support in the usual sup norm metric is not complete ? I know that this result can be proved by using the fact that the given space is dense in the space of all continuous functions that vanish at infinity which is complete , but I want a bit easier proof of this as I have not studied measure theory . I was trying to solve it directly from the definition of completeness.

Thanks for any help.

$\endgroup$

1 Answer 1

Reset to default
8
$\begingroup$

Take a particular example of a continuous function that goes to $0$ at $\pm \infty$, and a sequence of continuous functions of compact support that converges uniformly to it. This is a Cauchy sequence ...

$\endgroup$
6
  • $\begingroup$ Hello @Robert ;I took the function to be $f(x)=\dfrac{1}{1+x^2}$ but how to define the sequence that converges to $f$ ;Can you please help me $\endgroup$
    – Learnmore
    Commented Oct 14, 2016 at 14:02
  • 2
    $\begingroup$ For example, $f_n(x) = \dfrac{1-(x/n)^2}{1+x^2}$ for $-n \le x \le n$, $0$ otherwise. $\endgroup$
    – Robert Israel
    Commented Oct 14, 2016 at 15:25
  • $\begingroup$ @jaggu That's precisely why it should be $[-n,n]$. $\endgroup$
    – Robert Israel
    Commented Dec 28, 2016 at 16:57
  • $\begingroup$ @RobertIsrael......... oops, continuity, sorry $\endgroup$
    – Dhrubajyoti Sarkar
    Commented Dec 28, 2016 at 20:59
  • $\begingroup$ @RobertIsrael, do you know how to find the completion of this metric space in question? $\endgroup$
    – user561334
    Commented Dec 7, 2018 at 16:34

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .