## Linear Operators: General theory |

### From inside the book

Results 1-3 of 86

Page 421

It is obvious that y is a linear functional on the subspace T ( X ) of En . By II.3.11 , it can be extended to a linear functional 4. on E . By IV.3.7 , 41 has the form Wi [ Y1 , . . . , Yn ] ] = { diyi

It is obvious that y is a linear functional on the subspace T ( X ) of En . By II.3.11 , it can be extended to a linear functional 4. on E . By IV.3.7 , 41 has the form Wi [ Y1 , . . . , Yn ] ] = { diyi

**Hence**n i = 1 g ( x ) = 2a ( ) ...Page 423

Therefore , T'is weakly continuous at the origin , and

Therefore , T'is weakly continuous at the origin , and

**hence**at every point . Conversely , suppose that T is weakly continuous , and y * € Y * . Then y * T is a linear functional on X which is X * -continuous .**Hence**, by Theorem 9 ...Page 485

**Hence**T * is weakly compact . Conversely , if T * is weakly compact , it follows 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 ...### 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 |

quences | 26 |

Copyright | |

80 other sections not shown

### Other editions - View all

### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen 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