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August 19th, 2013, 12:37 PM  #1 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  Routine proof in Functional Analysis
This problem is quite routine, but I am not quite getting a finger on it. I need to prove that any selfadjoint linear map defined everywhere on a Hilbert space is bounded. I understand that either the closed graph theorem or uniform boundedness principle can be adequately applied, but I have blanked out. Please, any clearly stated proof is most welcome.

August 26th, 2013, 04:01 AM  #2 
Member Joined: Nov 2009 From: France Posts: 98 Thanks: 0  Re: Routine proof in Functional Analysis
Indeed, closed graph theorem is a good way. Let be a sequence in converging to and such that , . Then for each , . The LHS converges to while the RHS goes to .

August 26th, 2013, 11:50 AM  #3 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  Re: Routine proof in Functional Analysis
Yes, Girdav, I see you have indicated continuity at 0 which in turn gives us continuity everywhere. If you can obtain and post the proof using either the closed graph or uniform boundedness principle, I'll be more than grateful.

August 27th, 2013, 07:27 AM  #4 
Member Joined: Nov 2009 From: France Posts: 98 Thanks: 0  Re: Routine proof in Functional Analysis
AfroMike, I'm probably misunderstanding your request, but the sketch of proof did use closed graph theorem.

August 28th, 2013, 10:00 AM  #5 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  Re: Routine proof in Functional Analysis
We need to show that for any sequence in the graph of which converges where , then where is the self adjoint operator. You have established clearly that if then . For other the result is not so clear. This is the reasoning I have to prove using the closed graph theorem; perhaps you have a more concise approach?

August 28th, 2013, 11:33 AM  #6 
Member Joined: Nov 2009 From: France Posts: 98 Thanks: 0  Re: Routine proof in Functional Analysis
Let and . Define . Then by the case . Hence .

August 28th, 2013, 12:23 PM  #7 
Senior Member Joined: Dec 2012 Posts: 372 Thanks: 2  Re: Routine proof in Functional Analysis
Ok, I guess its also correct to make the conclusion that since and then . Under this assumption, we also have the desired result. 

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