By John B Conway

ISBN-10: 0387972455

ISBN-13: 9780387972459

This publication is an introductory textual content in practical research. in contrast to many sleek remedies, it starts off with the actual and works its strategy to the extra normal. From the reports: "This publication is a superb textual content for a primary graduate path in practical analysis....Many attention-grabbing and demanding purposes are included....It contains an abundance of routines, and is written within the attractive and lucid type which we have now come to anticipate from the author." --MATHEMATICAL stories

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C) 1::= 1 I (X,. I < oo . 19 §5. Isomorphic Hilbert Spaces and the Fourier Transform 1 3. A = V 8. A, show that Ph = "L { ( h, e ) e: eel} for every h in Jf. 14. Let A. ). Find II z" II , n � 0. )? • • • • • • 1 5. In the proof of (4. 14), show that if either e or '1 is finite, then e = '1· 16. If Jf is an infinite dimensional Hilbert space, show that no orthonormal basis for Jf is a Hamel basis. Show that a Hamel basis is uncountable. 1 7. l be a regular Borel measure on Rd. Show that L2(J,L) is separable.

Hence Eh = Ph by uniqueness. 7), I E I � 1 . But Eh = h for h in ran E, so I E I = 1 . l . Now ran(J - E) = ker E, so h - Eheker E. Hence 0 = ( h - Eh, h· ) = II h II 2 - ( Eh, h ) . l , II h - Eh I 2 = I h II 2 - 2 Re ( Eh, h ) + II Eh I 2 = 0. l ker(/ - E) = ran E. l . L. L and E is a projection. c: = = II. l . Hence ( Eh, h ) = ( E(h 1 + h 2 ), h 1 + h 2 ) = ( Eh 1 , h 1 ) = ( h 1 , h 1 ) = II h 1 11 2 � 0. (f) => (a): Let h 1 eran E and h 2 eker E. Then by (f), O � ( E(h 1 + h 2), h 1 + h 2 ) = ( h 1 , h 1 ) + ( h 1 , h 2 ) .

Cardinality. If Je is a Hilbert space, any two bases have the same PROOF. Let tC and � be two bases for Je and put e =the cardinality of tC, '1 =the cardinality of�- If e or '1 is finite, then e ='1 (Exercise 15). Suppose both e and '7 are infinite. Fore in tC, let �e = {f e �: (e, f) =F 0 } ; so �e is countable. 13b), each f in � belongs to at least one set �e' e in tC. • That is, � = u {�e: eetC }. Hence '1 � e · �0 =e. Ye. 15. Definition. The If (X, d) is a metric space that is separable and {Bi =B(x i; ei ): iei} is a collection of pairwise disjoint open balls in X, then I must be countable.

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A Course in Functional Analysis by John B Conway

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