## Linear Operators, Part 1 |

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Results 1-3 of 76

Page 89

Let X o y be the direct sum of the linear spaces X and Y in the sense of Section

1.11 , with the product topology of Section 1.8 . Then X Y is readily

topological linear space . If X and Y are B- ( or F- ) spaces , then X e Y is a B( or

an ...

Let X o y be the direct sum of the linear spaces X and Y in the sense of Section

1.11 , with the product topology of Section 1.8 . Then X Y is readily

**seen**to be atopological linear space . If X and Y are B- ( or F- ) spaces , then X e Y is a B( or

an ...

Page 181

By the argument used in the proof of Corollary 5 it is

to prove [ * ] in the case when g is a positive real function . It is also clear that [ * ]

holds for 2 - integrable simple functions . As in the proof of Theorem 4 , there is ...

By the argument used in the proof of Corollary 5 it is

**seen**that it will be sufficientto prove [ * ] in the case when g is a positive real function . It is also clear that [ * ]

holds for 2 - integrable simple functions . As in the proof of Theorem 4 , there is ...

Page 254

Thus by forming the chain Up , WQ * • . , Uq ' , Vp ' it is

UB and thus that vp is in V. Since { vp } is a basis , the vector w has an expansion

of the form Ug = Eb ( Uq ; vb ) VB , is in the closed linear manifold determined by

...

Thus by forming the chain Up , WQ * • . , Uq ' , Vp ' it is

**seen**that is equivalent toUB and thus that vp is in V. Since { vp } is a basis , the vector w has an expansion

of the form Ug = Eb ( Uq ; vb ) VB , is in the closed linear manifold determined by

...

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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