## 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 # ye Y and...

Page 513

( ii ) The

inverse .

( 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 ... ( iv ) U * is one - to - one if and only if the**range**of U is dense in Y . ( v ) If U maps onto y , then U * has a continuousinverse .

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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