## Linear Operators: General theory |

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Page 262

it is

following theorem is converse to this statement . 2 THEOREM . If S is normal ,

there is an isometric isomorphism between C * ( S ) and rba ( S ) such that

corresponding ...

it is

**clear**that the integral is a continuous linear functional on C ( S ) . Thefollowing theorem is converse to this statement . 2 THEOREM . If S is normal ,

there is an isometric isomorphism between C * ( S ) and rba ( S ) such that

corresponding ...

Page 282

It is

function f is said to be almost periodic if it is continuous and if for every e > 0 there

is an L = L ( € ) > 0 such that every interval in R of length L contains at least one ...

It is

**clear**that T ( $ ) CT ( d ) if € < d and that - te T ( 8 ) whenever t e T ( E ) . Thefunction f is said to be almost periodic if it is continuous and if for every e > 0 there

is an L = L ( € ) > 0 such that every interval in R of length L contains at least one ...

Page 292

Since it is

follows . Q . E . D . 9 THEOREM . A subset K of L ( S , E , u ) is weakly sequentially

compact if and only if it is bounded and the countable additivity of the integrals ...

Since it is

**clear**that Σ2 Σ3 , we conclude that Σ2Σ , from which the desired resultfollows . Q . E . D . 9 THEOREM . A subset K of L ( S , E , u ) is weakly sequentially

compact if and only if it is bounded and the countable additivity of the integrals ...

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### Contents

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Three Basic Principles of Linear Analysis | 49 |

Copyright | |

50 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

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