## Linear Operators: General theory |

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Page 487

Operators with Closed

the

= 0 implies y * y = 0 . Or , in other words , UX = { y \ U * y * = 0 ) implies y * y = 0 } ...

Operators with Closed

**Range**It was observed in Lemma 2 . 8 that the closure ofthe

**range**of an operator U € B ( X , Y ) consists of those vectors y such that y * UX= 0 implies y * y = 0 . Or , in other words , UX = { y \ U * y * = 0 ) implies y * y = 0 } ...

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 + y e Y and ...Page 514

19 If E is a projection with n dimensional

dimensional

e . , is bounded ) if and only if the

...

19 If E is a projection with n dimensional

**range**, then E * is a projection with ndimensional

**range**. ... 21 A linear mapping E such that E2 = E is a projection ( i .e . , is bounded ) if and only if the

**ranges**of E and I - E are closed . 22 Let E be a...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero