## Linear Operators: General theory |

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Page 151

Nelson Dunford, Jacob T. Schwartz. n - > 00 JE m , n - 00 m , n - > 00 . lt ( s ) P

vsu , ds ) < c and lim [ c \ / n ( 8 ) — f ( s ) | pv ( u , ds ) = 0 follow easily from

Corollary 13 ( b ) and Theorem 3 . 6 .

S . \ .

Nelson Dunford, Jacob T. Schwartz. n - > 00 JE m , n - 00 m , n - > 00 . lt ( s ) P

vsu , ds ) < c and lim [ c \ / n ( 8 ) — f ( s ) | pv ( u , ds ) = 0 follow easily from

Corollary 13 ( b ) and Theorem 3 . 6 .

**Consequently**, lim sup \ tn – Imlo Slim sup {S . \ .

Page 254

Thus { ua } and { vs } break up into a disjoint union of corresponding pairs U , V of

equivalence classes , each U having the same cardinality as the corresponding V

.

Thus { ua } and { vs } break up into a disjoint union of corresponding pairs U , V of

equivalence classes , each U having the same cardinality as the corresponding V

.

**Consequently**{ ua } and { v } have the same cardinality . Q . E . D . 15 ...Page 637

estimate for the series monoxin ) ( t ) which majorizes noo | Sn ( t ) ) . By Lemma

20 ( c ) , for each w > W , there exists a constant Mo < oo such that y ( t ) = \ PT ( t )

...

**Consequently**, ( i ) , . . . , ( v ) are proved inductively for all n . We now obtain anestimate for the series monoxin ) ( t ) which majorizes noo | Sn ( t ) ) . By Lemma

20 ( c ) , for each w > W , there exists a constant Mo < oo such that y ( t ) = \ PT ( t )

...

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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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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