Let be a Hilbert space and suppose . Then for all with .

please! ... I don't understand its.

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- Oct 1st 2011, 07:41 AMkonnaProve Opial's Property (Theorem) in Hilbert space
Let be a Hilbert space and suppose . Then for all with .

please! ... I don't understand its. - Oct 3rd 2011, 12:52 AMOpalgRe: Prove Opial's Property (Theorem) in Hilbert space
- Oct 3rd 2011, 06:05 AMkonnaRe: Prove Opial's Property (Theorem) in Hilbert space
Thank you so much. I think it is easily proof, when you show for me.

By the way, I have other proof of this theorem . Can you check it for me?

From Parallelogram Law, we have

That is

or --------(*)

Now take inferior limit as (*), we have

Since , we obtains that

It follow from above and our assumption ; , that

Thus, there is a positive integer such that , for all

Consider, for all

we can put

This implies that

or

Taking the inferior limit, we get

and so

Thus

It follows that and hence

#

I tried to do it. But not sure what to do.