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April 28th, 2011, 05:53 AM   #1
Joined: Aug 2009

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invertible linear operator not bounded below

Hello guys

In Murphys' "C-* Algebras and Operator Theory", Sect 1.4 page 21, it asserts that every invertible linear map between Banach spaces is bounded below, i.e.
there is a such that for all vectors.

That does now seems reasonable since I can for example take eigenvalues going to 0 but all positive. In , the operator

will not be bounded bellow but will be invertible with inverse given by


is there any errata of murphy's book? I've taken a little search on the internet and found none...

thank you guys!

EDIT: my example is stupid, the inverse is not bounded, but anyways, i will try to prove the assertion. thanks for sharing this moment with me
EDIT2: i feel even stupider, first because stupider must not even exist, and also because the proof is obvious taking uu^{-1} as bounded. next time i'll think before post, thanks for your cooperation
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April 28th, 2011, 08:33 AM   #2
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Re: invertible linear operator not bounded below


Don't berate yourself too harshly, I know the feeling. Especially with subjects like functional analysis, if you read for too long, all the definitions and properties start to blend together into a real mess. I've had many moments of the exact same kind, where all of a sudden I think something absurd to be true, and subsequently waste an hour to find my bearings...

Reminds me of this old math joke where the professor is at the board proving a theorem, and with regard to one assertion, he declares that "it is obviously so." But then he pauses, and begins to stare at the board, and mutters to himself "is it really obvious?", begins to pace back and forth in front of his formula-ridden chalkboard while his students look at him quizzically, and finally excuses himself with a few unintelligible words as he goes out into the hallway and walks up and down the hallway, down the stairs, back up, and about the entire department, all the while arguing with himself... after half an hour, as the students begin to lose patience and are about to leave, the professor bursts back into the room and declares loudly "it IS obvious!"

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