## Linear Operators: General theory |

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Page 6

... which proves the assertion ( v ) . It will next be

condition ( i ) is vacuously satisfied by P . To prove that P has the property ( ii ) ,

let X e P . It will be

...

... which proves the assertion ( v ) . It will next be

**shown**that P is admissible . Thecondition ( i ) is vacuously satisfied by P . To prove that P has the property ( ii ) ,

let X e P . It will be

**shown**that if z € A , and 2 < f ( x ) , then f ( x ) = f ( x ) . From ( v )...

Page 156

... are explicitly discussed in the next chapter , where it is

conjugate spaces of some familiar B - spaces may be represented in terms of set

functions . In the present section it will be

additive set ...

... are explicitly discussed in the next chapter , where it is

**shown**that theconjugate spaces of some familiar B - spaces may be represented in terms of set

functions . In the present section it will be

**shown**that the spaces of boundedadditive set ...

Page 335

It will first be

we let W , be a totally ordered subset of W ( 1 . 22 ) and let c CUW . . Then , for

some a e W . , cna is not void . Let æ be the smallest element of cna and let y be ...

It will first be

**shown**that W satisfies the hypothesis of Zorn ' s lemma . To do thiswe let W , be a totally ordered subset of W ( 1 . 22 ) and let c CUW . . Then , for

some a e W . , cna is not void . Let æ be the smallest element of cna and let y be ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero