## Linear Operators: General theory |

### From inside the book

Results 1-3 of 79

Page 598

Suppose that gn ( T ) converges in the

operator . Let € 0 ( T ) , and limn - 8n ( 2 ) + 0 . Show that 2 is a pole of o ( T ) , and

that E ( 2 ; T ) X has a positive finite dimension . ( Hint . See Exercise VII . 5 . 35 . )

...

Suppose that gn ( T ) converges in the

**uniform**operator topology to a compactoperator . Let € 0 ( T ) , and limn - 8n ( 2 ) + 0 . Show that 2 is a pole of o ( T ) , and

that E ( 2 ; T ) X has a positive finite dimension . ( Hint . See Exercise VII . 5 . 35 . )

...

Page 841

17 ( 23 ) representation as a C - space , almost periodic functions , IV . 7 . 6 ( 285

) bounded functions , IV . 6 . 18 - 22 ( 274 – 277 ) special C - spaces , ( 397 – 398

)

17 ( 23 ) representation as a C - space , almost periodic functions , IV . 7 . 6 ( 285

) bounded functions , IV . 6 . 18 - 22 ( 274 – 277 ) special C - spaces , ( 397 – 398

)

**uniform**continuity , 1 . 6 . 16 – 18 ( 23 – 24 ) of almost periodic functions , IV .Page 857

5 ( 32 ) remarks on , ( 730 )

properties , VI . 9 . 11 - 12 ( 512 - 513 ) Unit , of a group , ( 34 ) Unit sphere in a

normed space , compactness and finite dimensionality of , IV . 3 . 5 ( 245 ) ...

5 ( 32 ) remarks on , ( 730 )

**Uniform**operator topology , definition , VI . 1 . 1 ( 475 )properties , VI . 9 . 11 - 12 ( 512 - 513 ) Unit , of a group , ( 34 ) Unit sphere in a

normed space , compactness and finite dimensionality of , IV . 3 . 5 ( 245 ) ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

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### 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

analytic applied arbitrary assumed B-space Borel bounded called Chapter clear closed complex condition Consequently constant contains continuous functions continuous linear converges Corollary countably additive defined DEFINITION denote dense determined dimensional disjoint element equation equivalent everywhere Exercise exists extended field finite follows formula function defined function f given Hence Hilbert identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear map linear operator linear space meaning metric space neighborhood norm obtained operator positive measure space projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement strongly subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero