## Linear Operators: General theory |

### From inside the book

Results 1-3 of 39

Page 164

... on Eto E , is non - negative , it follows that { ūn ( E ) } is a bounded non -

decreasing set of real numbers for each Ee & . We define 2 ( E ) = limnīn ( E ) , E

€ En . By Corollary 4 , 2 , is countably additive on £1 , and hence its

£ is in ...

... on Eto E , is non - negative , it follows that { ūn ( E ) } is a bounded non -

decreasing set of real numbers for each Ee & . We define 2 ( E ) = limnīn ( E ) , E

€ En . By Corollary 4 , 2 , is countably additive on £1 , and hence its

**restriction**to£ is in ...

Page 166

In the following discussion of this other type of

assume that u is non - negative . Suppose that E is a set in £ . If we put E ( E ) = {

FeE | F C E } it is clear that E ( E ) is a field of subsets of E , and that E ( E ) is the ...

In the following discussion of this other type of

**restriction**it is not necessary toassume that u is non - negative . Suppose that E is a set in £ . If we put E ( E ) = {

FeE | F C E } it is clear that E ( E ) is a field of subsets of E , and that E ( E ) is the ...

Page 660

In Sections 5 , 6 , and 7 below , this

reader will see readily that a slight rewording of the proofs given ( if , indeed , any

is needed ) will establish the results of these sections for real B - spaces also .

In Sections 5 , 6 , and 7 below , this

**restriction**is not really essential , and thereader will see readily that a slight rewording of the proofs given ( if , indeed , any

is needed ) will establish the results of these sections for real B - spaces also .

### What people are saying - Write a review

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

### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 other sections not shown

### Other editions - View all

### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero