Let X be a open subset on Rn.

Let S be the set of all real, bounded, andfunctions on X, and rho is the sup norm.continuously diferentiable

Is (S,rho) a complete metric space?

Could you give me a counter-example if it is not?

Thanks,

JP

Printable View

- May 3rd 2010, 01:21 PMjaveiroComplete metric space
Let X be a open subset on Rn.

Let S be the set of all real, bounded, andfunctions on X, and rho is the sup norm.__continuously diferentiable__

Is (S,rho) a complete metric space?

Could you give me a counter-example if it is not?

Thanks,

JP - May 3rd 2010, 02:04 PMDrexel28
- May 3rd 2010, 03:20 PMjaveiro
Hum...

I guess it would be incomplete.

I was wondering if (abs(x))^(1+1/n) would be a counter example.

It converges to abs(x), but i'm failing to show that it is Cauchy... - May 3rd 2010, 04:23 PMDrexel28
What you said makes no sense. If you believe that this space is incomplete then you would want to find a Cauchy sequence that doesn't converge. Also, every Cauchy sequence is convergent.

But, let's assume you meant to take the open subset of to be and the function . Is that even differentiable at ?

I haven't checked it out, but what about with . This is differentiable and continuous, and I'm pretty sure Cauchy but it "wants" to converge to which is only differentiable at , not continuously differentiable.