## 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 € 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

and ...

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. Ux = PROOF . Let 0 #ye Yand ...

Page 514

19 If E is a projection with n dimensional

dimensional

, 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 ...### What people are saying - Write a review

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### Contents

Preliminary Concepts 1 1 Preliminary Concepts | 1 |

B Topological Preliminaries 10 B Topological Preliminaries | 10 |

Metric Spaces | 18 |

Copyright | |

80 other sections not shown

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