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August 19th, 2013, 12:41 PM   #1
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Routine proof in Functional Analysis

I have a rather routine problem, but cannot quite get a finger on it. I need to prove that any self-adjoint linear map defined everywhere on a Hilbert space is bounded. I understand that either the closed graph theorem, or the uniform boundedness principle may be adequately applied; but I have blanked out. Please any clearly stated proof is more than welcome.
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August 21st, 2013, 10:57 AM   #2
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Re: Routine proof in Functional Analysis

Ok, so I have deviced a proof using neither of the two theorems. Given such that for all y in H with being linear and self-adjoint, we will verify that is continuous at zero. With this result we have also that is continuous at zero, giving us continuity of A on H; as a consequence of A being linear. It suffices to check that for any sequence in H which converges weakly to zero, we have also converging weakly to A(0) = 0. Given that converges weakly to zero, then for any element , we have that converges to T(0) = 0. In particular, for ,

, using the self-adjoint property of A.

This is to say that converges to 0 in the topology of which coincides with the weak topology on since H is a Hilbert space. As such, we have the desired result using the weak topology on the domain and co-domain, which in turn gives us continuity in the normed sense as initially proposed.

I am not satisfied though. I still hope for some analysis gurus to someday give proofs using the closed graph theorem and uniform boundedness principle.
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