## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 89

Page 54

A linear mapping of one F - space into another is continuous if and only if it maps

be linear and continuous , and let B CX be

...

A linear mapping of one F - space into another is continuous if and only if it maps

**bounded**sets into**bounded**sets . PROOF . Let X , Y be F - spaces , let T : X + Ybe linear and continuous , and let B CX be

**bounded**. For every neighborhood V...

Page 97

and to be

function u is important because it dominates u in the sense that v ( u , E ) > N ( E )

for E € £ ; the reader should test his comprehension of Definition 4 below by ...

and to be

**bounded**if u is**bounded**. The total variation v ( u ) of an additive setfunction u is important because it dominates u in the sense that v ( u , E ) > N ( E )

for E € £ ; the reader should test his comprehension of Definition 4 below by ...

Page 345

if and only if it is

and each j = 0 , 1 , . . . , p . ( c ) Co is not weakly complete , and not reflexive . ( d )

A subset A CCP is conditionally compact if and only if it is

...

if and only if it is

**bounded**and prin ( s ) converges ( to f ( s ) ) for each s in [ a , b ]and each j = 0 , 1 , . . . , p . ( c ) Co is not weakly complete , and not reflexive . ( d )

A subset A CCP is conditionally compact if and only if it is

**bounded**and for each...

### 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