## Linear Operators: General theory |

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Page 161

Hence Plin → M , which proves that

therefore , that

numbers , then according to Lemma 1 . 5 , sup lu ( E ) = v ( u , S ) < 4 sup su ( E ) )

. EEE This ...

Hence Plin → M , which proves that

**ba**(**S**, E , X ) is complete . It follows ,therefore , that

**ba**(**S**, E , X ) is a B - space . If X is the set of real or complexnumbers , then according to Lemma 1 . 5 , sup lu ( E ) = v ( u , S ) < 4 sup su ( E ) )

. EEE This ...

Page 311

Q . E . D . Next we turn to an investigation of the space

The space

) is also weakly complete . PROOF . Consider the closed subspace B ( S , E ) of ...

Q . E . D . Next we turn to an investigation of the space

**ba**(**S**, E ) . 9 THEOREM .The space

**ba**(**S**, E ) is weakly complete . If S is a topological space , the rba ( S) is also weakly complete . PROOF . Consider the closed subspace B ( S , E ) of ...

Page 340

16 Let S be a completely regular topological space . Show that C ( S ) is

separable if and only if S is compact and metric . 17 Show that a sequence { an }

of elements of

only if ...

16 Let S be a completely regular topological space . Show that C ( S ) is

separable if and only if S is compact and metric . 17 Show that a sequence { an }

of elements of

**ba**(**S**, E ) converge weakly to an element à e**ba**(**S**, E ) if andonly if ...

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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