## Linear Operators: General theory |

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Results 1-3 of 47

Page 373

44 ] and for integrals by F .

; p . 36 ] and was also used by Sz . - Nagy [ 5 ] . A similar argument may be

applied to obtain this result in any uniformly convex B - space . This generalizes

and ...

44 ] and for integrals by F .

**Riesz**[ 2 ; p . 456 ] . Lemma 4 . 2 is due to F .**Riesz**[ 8; p . 36 ] and was also used by Sz . - Nagy [ 5 ] . A similar argument may be

applied to obtain this result in any uniformly convex B - space . This generalizes

and ...

Page 388

1363 ] and F .

converges strongly to | in L ( S , E , u ) if and only if it converges weakly and in i .

This theorem remains valid in any uniformly convex B - space . Theorem 8 . 15

was ...

1363 ] and F .

**Riesz**[ 13 ] . THEOREM . If 1 < p < oo then a sequence { In }converges strongly to | in L ( S , E , u ) if and only if it converges weakly and in i .

This theorem remains valid in any uniformly convex B - space . Theorem 8 . 15

was ...

Page 608

However , it seems fair to state that it was F .

developed it along the lines presented here . The reader of his book ( F .

3 ) , particularly Secs . 71 - 82 ) will find many of these concepts and results ...

However , it seems fair to state that it was F .

**Riesz**who most fully revealed anddeveloped it along the lines presented here . The reader of his book ( F .

**Riesz**(3 ) , particularly Secs . 71 - 82 ) will find many of these concepts and results ...

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### Contents

Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |

Partially Ordered Systems | 7 |

Exercises | 9 |

Copyright | |

35 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

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