## Linear Operators, Part 1 |

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Page 245

It was

upon x . Thus if U is a linear operator on X it is seen that Uw = Q Ubit . . . tanUbn

is also continuous in x . Q . E . D . 5 THEOREM . A normed linear space is finite ...

It was

**shown**in the proof of Lemma 1 that dig i = 1 , . . . , n , depends continuouslyupon x . Thus if U is a linear operator on X it is seen that Uw = Q Ubit . . . tanUbn

is also continuous in x . Q . E . D . 5 THEOREM . A normed linear space is finite ...

Page 335

It will first be

we let W , be a totally ordered subset of W ( I . 22 ) and let c CuW . Then , for

some a € Wocna is not void . Let æ be the smallest element of cha and let y be

any ...

It will first be

**shown**that W satisfies the hypothesis of Zorn ' s lemma . To do thiswe let W , be a totally ordered subset of W ( I . 22 ) and let c CuW . Then , for

some a € Wocna is not void . Let æ be the smallest element of cha and let y be

any ...

Page 479

It only remains to be

* in Y * with y * = 0 , y * T = T * y * = 0 . This contradicts the assumption that 7 * is

one - to - one , and proves the lemma . Q . E . D . The final argument in the ...

It only remains to be

**shown**that TX = Y . If ye Y and y¢ TX , there is ( II . 3 . 13 ) a y* in Y * with y * = 0 , y * T = T * y * = 0 . This contradicts the assumption that 7 * is

one - to - one , and proves the lemma . Q . E . D . The final argument in the ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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

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