## Linear Operators, Part 1 |

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Page 120

PROOF . The function y ( t ) = tP / p + t - 9 / 9 has a positive derivative for t > 1 ,

and a negative derivative for 0 < t < 1 . Hence , its minimum value for t > 0 is 9 ( 1

) = 1 . If we put t = alla b - 1 / p we obtain the

a ...

PROOF . The function y ( t ) = tP / p + t - 9 / 9 has a positive derivative for t > 1 ,

and a negative derivative for 0 < t < 1 . Hence , its minimum value for t > 0 is 9 ( 1

) = 1 . If we put t = alla b - 1 / p we obtain the

**inequality**ab < app + 6° / q , valid fora ...

Page 121

In the sequel the symbol | rather than [ / ] will be used for an element in Lp . We

observe that the

functions ) the

.

In the sequel the symbol | rather than [ / ] will be used for an element in Lp . 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 Lo.

Page 248

The above

follows from the postulates for H that the Schwarz

is zero . Hence suppose that + 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 + 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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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero