## Linear Operators: General theory |

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Page 487

It follows from Theorem IV.6.7 , that T ( S ) is conditionally compact in the metric of

Y if and only if the condition is satisfied . Q.E.D. 6. Operators with Closed

was observed in Lemma 2.8 that the closure of the

It follows from Theorem IV.6.7 , that T ( S ) is conditionally compact in the metric of

Y if and only if the condition is satisfied . Q.E.D. 6. Operators with Closed

**Range**Itwas observed in Lemma 2.8 that the closure of the

**range**of an operator U e B ...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

define ...

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 anddefine ...

Page 489

since the

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

the previous lemma that U x = UX 3 . Hence , U has a closed

since the

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

**range**of U * is also closed . It follows fromthe previous lemma that U x = UX 3 . Hence , U has a closed

**range**. Q.E.D. 5 ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

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