## 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 513

( ii ) The

...

( 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 = Tx such that Kyl . ( iii ) U is one - to - one if the**range**of U * is dense in X * . ( iv ) U * is one - to - one if and only if the**range**of U...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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