## Linear Operators: General theory |

### From inside the book

Results 1-3 of 80

Page 421

7 , y , has the form Valyv , . . , Yn ] = { Xi Yi

. E . D . PROOF OF THEOREM 9 . Every functional in l ' is - continuous , by

Lemma 8 . Conversely , let g + 0 be a linear functional on X which is I ' -

continuous .

7 , y , has the form Valyv , . . , Yn ] = { Xi Yi

**Hence**g ( x ) = { vif : ( a ) , X e X . i = 1 Q. E . D . PROOF OF THEOREM 9 . Every functional in l ' is - continuous , by

Lemma 8 . Conversely , let g + 0 be a linear functional on X which is I ' -

continuous .

Page 441

Put K ; = colqi + U ) OQ ) C9i + U . Then Ki is a closed , and

subset of co ( Q ) .

an easy induction on Lemma 2 . 5 . It follows readily that p has the form p = _ , a ;

kı ...

Put K ; = colqi + U ) OQ ) C9i + U . Then Ki is a closed , and

**hence**a compact ,subset of co ( Q ) .

**Hence**CO ( Q ) = co ( K , U . . . UK , ) = co ( K , U . . . UKn ) , byan easy induction on Lemma 2 . 5 . It follows readily that p has the form p = _ , a ;

kı ...

Page 485

from Lemma 7 that T * * is continuous relative to the X * , Y * * * topologies in X * *

, Y * * , respectively . If S , S * * are the closed unit spheres in X , X * * ,

respectively ...

**Hence**7 * is weakly compact . Conversely , if T * is weakly compact , it followsfrom Lemma 7 that T * * is continuous relative to the X * , Y * * * topologies in X * *

, Y * * , respectively . If S , S * * are the closed unit spheres in X , X * * ,

respectively ...

### 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