## Linear Operators, Part 1 |

### From inside the book

Results 1-3 of 67

Page 488

It follows from the definition of U * that every element in its

stated condition . Q.E.D. - 3 LEMMA . If the adjoint of an operator U in B ( X , Y ) is

one - to - one and has a closed

It follows from the definition of U * that every element in its

**range**satisfies thestated condition . Q.E.D. - 3 LEMMA . If the adjoint of an operator U in B ( X , Y ) is

one - to - one and has a closed

**range**, then UX = Y. PROOF . Let 0 #ye Y and ...Page 489

since the

restriction of y * to 3 , then x * = U ** . Hence , the

follows from the previous lemma that U_X = UX 3 . Hence , U has a closed

.

since the

**range**of U * is closed , æ * U * y * for some y * e Y * . If z * is therestriction of y * to 3 , then x * = U ** . Hence , the

**range**of U * is also closed . Itfollows from the previous lemma that U_X = UX 3 . Hence , U has a closed

**range**.

Page 513

( ii ) The

the

of ...

( ii ) The

**range**of U is closed if there exists a constant K such that for any y in the**range**there exists a solution of y Tæ such that 2 SK y . ( iii ) U is one - to - one ifthe

**range**of U * is dense in X * . ( iv ) U * is one - to - one if and only if the**range**of ...

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero