## 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

, 13131 Tx - 1x T 7 ( 3 ) < Mel . 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

, 13131 Tx - 1x T 7 ( 3 ) < Mel . 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

To each point pe C there corresponds a unique nearest point N ( p ) € K. To see

this , note that Lemma IV.4.2

+ p - kil = inf zer \ p - k | then { k ; } converges , say , to qe K. If { k } } is another ...

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 lim ;+ p - kil = inf zer \ p - k | then { k ; } converges , say , to qe K. If { k } } is another ...

### What people are saying - Write a review

User Review - Flag as inappropriate

i want to read

### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

80 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 condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad 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 operator positive measure problem Proc PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero