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By Riaz A Usmani

ISBN-10: 0585158231

ISBN-13: 9780585158235

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If Proof. (a) a sequence (I Iy II} Consider the case where each (y } where n is unbounded. ) r or e G and x ¿ G . Consider n r r n=l,2,.... We claim that the sequence x xy =1, n n For if ||y || £ K I - xy where x THEOREM Let (a) x It is clear that if for any contrary to n = (x - x)y n n n for each n, then 0. sufficiently large. This provides w^ x i G . r Without loss of generality we may (taking a subsequence if necessary) -I We see that ^ «*. Set Z = M y I I suppose that n ' n' ' n XZn= IIyJ 1-1 0.

E. In particular, when n = 0, we get | | tx| | > k | | [ x ] | | which implies that the map from X/N(T) into Y has a continuous inverse. ■ T is an operator in B(X) with X/R(T) finite dimensional, then is closed. Proof. B LEMMA Let T be an operator in there exist bounded operators B(X). T^ Then and T^ T is Fredholm if and only if and compact operators and such that Proof. Suppose T e Ф(Х). T^T = I + (1) TT^ = I + ( 2) Then we can find closed subspaces X_ and such that X = N(T) Ф X^ = R(T) Ф X^. Now T restricted to X^ is an invertible operator; let T denote its X 38 inverse (defined on R(T)).

On the and the corresponding function From our preliminary discussion function, and is bounded as theorem approaches Thus (Xx) * -^- I all X |ф(Х)| ^ Clearly 0. ||x*|| Ф(Х) | |(Xx) * | |. ф(0) = 0 But then so that X = O is an entire By Liouville*s x*(x’) = 0 for which is impossible.! An immediate consequence is the following important result of Gelfand and Mazur. 2) THEOREM A normed division algebra over the complex field is isomorphic to the complex number system. 24 Proof. X - X Take x e A. 1 there exists a complex is not regular.

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Applied linear algebra by Riaz A Usmani

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