## Linear Operators: General theory |

### From inside the book

Results 1-3 of 49

Page 26

Hence the

- * 7 . Convergence and Uniform Convergence of Generalized Sequences The

notion of convergence introduced in Definition 6 . 5 is not sufficiently general for ...

Hence the

**identity**is a homeomorphism and the space is metrizable . Q . E . D . w- * 7 . Convergence and Uniform Convergence of Generalized Sequences The

notion of convergence introduced in Definition 6 . 5 is not sufficiently general for ...

Page 479

... if and only if its adjoint T * has a bounded inverse ( T * ) - 1 defined on all of X *

. When these inverses exist , ( T - 1 ) * = ( T * ) - 1 PROOF . If T - 1 exists and is in

B ( Y , X ) , then , by Lemma 4 , ( TT - 1 ) * = ( T - 1 ) * T * is the

... if and only if its adjoint T * has a bounded inverse ( T * ) - 1 defined on all of X *

. When these inverses exist , ( T - 1 ) * = ( T * ) - 1 PROOF . If T - 1 exists and is in

B ( Y , X ) , then , by Lemma 4 , ( TT - 1 ) * = ( T - 1 ) * T * is the

**identity**in Y * and ...Page 661

The

bounded and hence , by II . 1 . 18 , the set of x for which Tnx / n → 0 is a closed

linear manifold . Thus X , is a closed linear manifold and , since a continuous

linear ...

The

**identity**T ' n ( * ) - = A ( n ) - n - 1 - n A ( n - 1 ) n shows that { T " \ n } isbounded and hence , by II . 1 . 18 , the set of x for which Tnx / n → 0 is a closed

linear manifold . Thus X , is a closed linear manifold and , since a continuous

linear ...

### 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