## Linear Operators: General theory |

### From inside the book

Results 1-3 of 87

Page 60

Statement ( iv ) clearly

i ) , ( ii ) , and ( iv ) are equivalent . If M = sup ( Tx ] is finite , then for an arbitrary x

+ 0 , 12 31 \ Tx ] = v1 TG ) SMļa . This shows that ( iii )

Statement ( iv ) clearly

**implies**the continuity of T at 0 ; so ( iv )**implies**( ii ) . This (i ) , ( ii ) , and ( iv ) are equivalent . If M = sup ( Tx ] is finite , then for an arbitrary x

+ 0 , 12 31 \ Tx ] = v1 TG ) SMļa . This shows that ( iii )

**implies**( iv ) . It is obvious ...Page 280

That ( 1 )

14 to show that condition ( 3 ) of that theorem

follows that S may be embedded as a dense subset of a compact Hausdorff

space ...

That ( 1 )

**implies**( 2 ) can be proved in a manner similar to that used in Theorem14 to show that condition ( 3 ) of that theorem

**implies**( 4 ) . From Corollary 19 itfollows that S may be embedded as a dense subset of a compact Hausdorff

space ...

Page 454

Nelson Dunford, Jacob T. Schwartz, William G. Bade. Proof . To each point pe C

there corresponds a unique nearest point N ( p ) € K . To see this , note that

Lemma IV . 4 . 2

zek ...

Nelson Dunford, Jacob T. Schwartz, William G. Bade. Proof . To each point pe C

there corresponds a unique nearest point N ( p ) € K . To see this , note that

Lemma IV . 4 . 2

**implies**that if { k ; } is a sequence in K such that limi * \ p - k ; = infzek ...

### What people are saying - Write a review

User Review - Flag as inappropriate

i want to read

### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

10 other sections not shown

### Other editions - View all

### Common terms and phrases

Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements 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 mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero