By Bela Sz. -Nagy

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**Additional resources for Appendix to Frigyes Riesz and Bela Sz. -Nagy Functional Analysis... **

**Example text**

Thus F( 0) = Cl + C2 cos 2 0 for some constants Cl, C2. 9) we have for any f E

For any trace class operator T and any bounded operator X in 'Je let (T, X) = trTX. ~oo(ilf)' T E :JI('Je); (ii) The Banach space 'Zl3('Je) of all bounded operators in 'Je with the operator norm is isometrically isomorphic to the dual of :J I('Je) with the norm I . III under the correspondence X --+ C,X)IJ1(ilf)' X E 'Zl3('Je). 16. 18: Let 'Je be a Hilbert space of dimension ;::: 3. P('Je). L is one to one. P('Je). L = LjPjluj)(ujl. 8) is obtained. P('Je), tr pP = tr pp2 = tr P pP ;::: 0 since P pP ;::: O.

It follows that for n = 2,3, .... j Aj. 6: (Hahn-Hellinger Theorem [53]) Let (O,~) be a measurable space and let ~ be a O-valued observable in the Hilbert space cae. Then there exists a sequence AIX" AI, A2, ... 6) j-fold If A:x" A~, A~, . is another sequence of mutually singular O'-finite measures satisfying the same conditions then Aj is equivalent to Aj for each j = 00, 1,2, ... 5 there exist O'-finite measures VI » V2 » .. such that ~ '" ~Vl ffi ~V2 ffi ... Choose and fix a version of the Radon-Nykodym derivatives ¢n = ddvVnn- l ,n = 2,3, ....