## Linear Operators: General theory |

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This

This

**proves**the statement made above . We have seen that the mapping x → H ( x ) is a homomorphism . To show that it is an isomorphism , let X , # yo . We will**prove**that there exists an ho c H with ho ( x , ) #ho ( yo ) .Page 495

To

To

**prove**the theorem it will , by Theorem 3 , suffice to show that u ( E ) is in 2 ( X ) for every Borel set E. Let B be the Borel sets in S. Then the space B ( S , B ) ( cf. IV.2.12 ) is the space of bounded Borel measurable functions ...Page 654

7

7

**Prove**that if t > 0 and x € D ( An ) , then 1 T ( t ) x Σ Ak x + ( t - 8 " -1T ( $ ) A " x ds . k ! k = 0 ( n - 1 ) !. 8**Prove**that if x € D ( A " ) , then n - 1 tk lim tín | T ( t ) = - 2 Ak x t- > 0 k ! n !### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

91 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed closure complex condition Consequently contains continuous functions converges Corollary defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function f given Hence Hilbert space implies integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space linear topological space Math means measure space metric space neighborhood norm open set operator problem Proc projection Proof properties proved range reflexive respect Russian 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