My Math Forum invertible linear operator not bounded below

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 April 28th, 2011, 05:53 AM #1 Newbie   Joined: Aug 2009 Posts: 16 Thanks: 0 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 $\delta > 0$ such that $\| u(x) \| > \delta \|x\|$ for all vectors. That does now seems reasonable since I can for example take eigenvalues going to 0 but all positive. In $l^\infty$, the operator $u ( a_1, a_2,a_3, \ldots)= (\frac{a_1}{1}, \frac{a_2}{2} , \frac{a_3}{3}, \ldots)$ will not be bounded bellow but will be invertible with inverse given by $u^{-1} ( a_1,a_2,a_2, \ldots)= (a_1, 2 a_2 , 3 a_3 , \ldots)$ right? 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
 April 28th, 2011, 08:33 AM #2 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 Re: invertible linear operator not bounded below otaniyul, 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!" -Ormkärr-

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