Prove Opial's Property (Theorem) in Hilbert space

**konna** · October 1st 2011, 06:41 AM

Let $H$ be a Hilbert space and suppose $x_{n} \rightharpoonup x$ . Then $\liminf_{n\rightarrow\infty}||x_n-x|| < \liminf_{n\rightarrow\infty}||x_n-y||,$ for all $y \in H$ with $x \neq y$ .

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

**Opalg** · October 2nd 2011, 11:52 PM

Originally Posted by konna

Let $H$ be a Hilbert space and suppose $x_{n} \rightharpoonup x$ . Then $\liminf_{n\rightarrow\infty}||x_n-x|| < \liminf_{n\rightarrow\infty}||x_n-y||,$ for all $y \in H$ with $x \neq y$ .

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

Presumably the notation $x_{n} \rightharpoonup x$ means that $x_n$ converges weakly to $x.$ Then

$\|x_n-y\|^2 = \|(x_n-x) + (x-y)\|^2 = \|x_n-x\|^2 + 2\text{Re}\langle x_n-x, x-y \rangle + \|x-y\|^2.$

Now take the liminf, noting that the inner product term goes to 0 because of the weak convergence.

**konna** · October 3rd 2011, 05:05 AM

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

$2||x_n-x||^2+2||x-y||^2=||x_n-y||^2+||x_n-x-(x-y)||^2.$

That is $2||x_n-x||^2+2||x-y||^2=||x_n-y||^2+||x_n-x||^2-2\langle x_n-x,x-y\rangle+||x-y||^2,$

or $||x_n-y||^2-||x_n-x||^2=2 \langle x_n-x,x-y\rangle+||x-y||^2.$ --------(*)

Now take inferior limit as (*), we have

$\liminf_{n \rightarrow \infty}(||x_n-y||^2-||x_n-x||^2)=2\liminf_{n \longrightarrow \infty}\langle x_n-x,x-y\rangle+\liminf_{n \longrightarrow \infty}||x-y||^2.$

Since $x_n \rightharpoonup x$ , we obtains that $\liminf_{n \longrightarrow \infty}\langle x_n-x,x-y\rangle = 0.$

It follow from above and our assumption ; $x \neq y$ , that $\liminf_{n \longrightarrow \infty}(||x_n-y||^2-||x_n-x||^2)> 0.$

Thus, there is a positive integer $N$ such that $||x_n-y||^2-||x_n-x||^2> 0$ , for all $n \geq N.$

Consider, for all $n \geq N, (||x_n-y||-||x_n-x||)(||x_n-y||+||x_n-x||)> 0,$
we can put $M=\sup_{n \geq N}\{||x_n-y||+||x_n-x||\}.$

This implies that $M(||x_n-y||+||x_n-x||) \geq 2 \langle x_n-x,x-y\rangle+||x-y||^2,$

or $M||x_n-y|| \geq M||x_n-x||+ 2 \langle x_n-x,x-y\rangle+||x-y||^2$

Taking the inferior limit, we get

$M\liminf_{n \longrightarrow \infty}||x_n-y|| \geq M\liminf_{n \longrightarrow \infty}||x_n-x||+ 2 \liminf_{n \longrightarrow \infty}\langle x_n-x,x-y\rangle+||x-y||^2,$

and so $M\liminf_{n \longrightarrow \infty}||x_n-y|| \geq M\liminf_{n \longrightarrow \infty}||x_n-x||+||x-y||^2.$

Thus $M(\liminf_{n \longrightarrow \infty}||x_n-y|| - \liminf_{n \longrightarrow \infty}||x_n-x||) \geq ||x-y||^2 > 0.$

It follows that $\liminf_{n \longrightarrow \infty}||x_n-y|| - \liminf_{n \longrightarrow \infty}||x_n-x|| > 0,$ and hence

$\liminf_{n \longrightarrow \infty}||x_n-y|| > \liminf_{n \longrightarrow \infty}||x_n-x||.$ #

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

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