## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 83

Page 162

Thus ca ( S , E ) is the space of all real or complex valued countably additive set

functions defined on Σ . We have just

E ) are Bspaces . For some purposes it is useful to know that they are also ...

Thus ca ( S , E ) is the space of all real or complex valued countably additive set

functions defined on Σ . We have just

**seen**that the spaces ba ( S , E ) and ca ( S ,E ) are Bspaces . For some purposes it is useful to know that they are also ...

Page 254

Thus by forming the chain VB , UQ , . . . , Uq ' , Vg , it is

to vg , and thus that Vg is in V . Since { vs } is a basis , the vector u , has an

expansion of the form un = E ( UX , UB ) 0g , so that we is in the closed linear

manifold ...

Thus by forming the chain VB , UQ , . . . , Uq ' , Vg , it is

**seen**that vg is equivalentto vg , and thus that Vg is in V . Since { vs } is a basis , the vector u , has an

expansion of the form un = E ( UX , UB ) 0g , so that we is in the closed linear

manifold ...

Page 567

We have

) . Hence d ( 2 ) 2 | R ( 2 ; T ) | - 1 , from which the statements follow . Q . E . D . + 4

LEMMA . The closed set o ( T ) is bounded and non - void . Moreover sup lo ( T ) ...

We have

**seen**in the proof of Lemma 2 that if lul < | R ( 2 ; T ) - 1 , then 2 + ue o ( T) . Hence d ( 2 ) 2 | R ( 2 ; T ) | - 1 , from which the statements follow . Q . E . D . + 4

LEMMA . The closed set o ( T ) is bounded and non - void . Moreover sup lo ( T ) ...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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

Akad 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 isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric 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