True/false : The Space of all continiuos real valued functions with compact support with supnorm metric is complete .

Question

Is the followimg statement is True/false

The Space of all continiuos real valued functions with compact support with supnorm metric is complete . (True/false)

i have found the answer here : are they complete metric spaces?

Now my confusion is that i didn't understand the answer How can we contradicts this function ?

@KaviRamaMurthy No sir , Im preparing For NBHM exam it is a NBHM question — jasmine, Aug 05 '19 at 23:55

Kavi Rama Murthy · Accepted Answer · 2019-08-05T23:52:33.740

1

Hints: Each $f_n$ has compact support since $x^{2}>n-1$ implies $f_n(x)=0$, Verify that $f_n(x) \to f(x) \equiv \frac 1 {1+x^{2}}$ uniformly on $\mathbb R$. Conclude that $\{f_n\}$ is Cauchy in the given space. Suppose it converges to some $g$ in the given space. Then $f_n \to f$ and $f_n \to g$ pointwise. Hence $f(x)=g(x)$ for all $x$. But $g$ has compact support and $f$ doesn't. This completes the proof.

edited Aug 05 '19 at 23:52

answered Aug 05 '19 at 23:45

Kavi Rama Murthy

311,013

Im not getting How $ f$ does not have compact support i mean u have already take that each $f_n $ has compact support – jasmine Aug 05 '19 at 23:51
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$f(x)=\frac 1 {1+x^{2}} \neq 0$ for any $x$. – Kavi Rama Murthy Aug 05 '19 at 23:53
Got its sir thanks u sir @Kavi rama sir – jasmine Aug 05 '19 at 23:55

True/false : The Space of all continiuos real valued functions with compact support with supnorm metric is complete .

1 Answers1