## Linear Operators: General theory |

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Page 120

Sst ( s ) g ( s ) u ( ds ) < Wolle . Proof . The function y ( t ) = { " / p + t- / q has a

positive derivative for t > 1 , and a negative derivative for 0 < t < 1. Hence , its

minimum value for t > 0 is q ( 1 ) = 1. If we put t = alla b - 1 / P we obtain the

Sst ( s ) g ( s ) u ( ds ) < Wolle . Proof . The function y ( t ) = { " / p + t- / q has a

positive derivative for t > 1 , and a negative derivative for 0 < t < 1. Hence , its

minimum value for t > 0 is q ( 1 ) = 1. If we put t = alla b - 1 / P we obtain the

**inequality**ab ...Page 121

... We observe that the

functions ) the

. This observation is obvious in the case of Minkowski's

... We observe that the

**inequality**of Minkowski and ( in the case of scalar valuedfunctions ) the

**inequality**of Hölder may be regarded as applying to the spaces Lp. This observation is obvious in the case of Minkowski's

**inequality**. To see that it ...Page 248

The above

follows from the postulates for H that the Schwarz

is zero . Hence suppose that 8C # 0 #y . For an arbitrary complex number a 0 = (

x ...

The above

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

**inequality**is valid if either x or yis zero . Hence suppose that 8C # 0 #y . For an arbitrary complex number a 0 = (

x ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero