## Linear Operators, Part 1 |

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Page 289

fe Lo

to Lq , so that there is a functional y * e L * such that *** ( ** ) = y * ( 8 ) when g

and x * are connected , as in Theorem 1 , by the formula x * 1 = $ st ( s ) g ( s ) u (

ds ) ...

fe Lo

**Proof**. Let x ** € ( L % ) * . By Theorem 1 , L TM is isometrically isomorphicto Lq , so that there is a functional y * e L * such that *** ( ** ) = y * ( 8 ) when g

and x * are connected , as in Theorem 1 , by the formula x * 1 = $ st ( s ) g ( s ) u (

ds ) ...

Page 415

Hence ko e p - A , and thus pe A + k , CA + K . Q.E.D. Since the commutativity of

the group G is not essential to the

topological groups . 4 LEMMA . For arbitrary sets A , B in a linear space X : ( i ) co

...

Hence ko e p - A , and thus pe A + k , CA + K . Q.E.D. Since the commutativity of

the group G is not essential to the

**proof**, the same result holds for non - Abeliantopological groups . 4 LEMMA . For arbitrary sets A , B in a linear space X : ( i ) co

...

Page 699

The

11 as outlined in the diagram : C - CPx + DP , → DR → Cx . The

implication CPx DPX which is the

...

The

**proof**of this lemma is the most involved of all the steps in the**proof**of Lemma11 as outlined in the diagram : C - CPx + DP , → DR → Cx . The

**proof**of theimplication CPx DPX which is the

**proof**of Lemma 14 is very similar to the**proof**of...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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