## Linear Operators: Spectral theory |

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Page 1061

Thus ( HT sup SH1 ( ) g ( x ) dx = Kellos by Theorem IV.8.1 , the Hahn - Banach

theorem ( II.3.14 ) and Hölder's

< p < 00. Q.E.D. Having proved the basic

...

Thus ( HT sup SH1 ( ) g ( x ) dx = Kellos by Theorem IV.8.1 , the Hahn - Banach

theorem ( II.3.14 ) and Hölder's

**inequality**, and the theorem is proved for all p , 1< p < 00. Q.E.D. Having proved the basic

**inequality**of M. Riesz , we now proceed...

Page 1105

We now pause to sharpen another of the

continuity of the product TS which was noted in the paragraph following Lemma 9

, the continuity of the norm function which follows from the triangle

Lemma ...

We now pause to sharpen another of the

**inequalities**of Lemma 9 . ... thecontinuity of the product TS which was noted in the paragraph following Lemma 9

, the continuity of the norm function which follows from the triangle

**inequality**ofLemma ...

Page 1774

The above

follows from the postulates for Ý that the Schwarz

is zero . Hence suppose that x 70 #y . For an arbitrary complex number a 0 S ( x +

...

The above

**inequality**, known as the Schwarz**inequality**, will be proved first . Itfollows from the postulates for Ý that the Schwarz

**inequality**is valid if either x or yis zero . Hence suppose that x 70 #y . For an arbitrary complex number a 0 S ( x +

...

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