## Linear Operators: General theory |

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Page 177

1 - 00 nE may be

represented as the difference of its positive and negative variations ( 4 . 11 ) and

so we may also

1 - 00 nE may be

**assumed**that 2 is real valued . A real valued set function can berepresented as the difference of its positive and negative variations ( 4 . 11 ) and

so we may also

**assume**that 2 is positive . Let , then , { En } be a sequence of ...Page 278

Let H be an algebraic homomorphism of C ( S ) into C ( T ) , where S and T are

compact Hausdorff spaces . If the algebras C ( S ) and C ( T ) are over the field of

complex numbers , it is also

has ...

Let H be an algebraic homomorphism of C ( S ) into C ( T ) , where S and T are

compact Hausdorff spaces . If the algebras C ( S ) and C ( T ) are over the field of

complex numbers , it is also

**assumed**that H1 = Hf . Then H is continuous andhas ...

Page 675

We may and shall

( s ) 2 1 } . Since f € L , it follows that ula ) < oo and thus f is the sum of a

summable function fXa and a bounded function txa » . Now , since Tlo S1 , we

have \ A ...

We may and shall

**assume**that | 2 0 . Let a ' be the complement of the set a = { s \ /( s ) 2 1 } . Since f € L , it follows that ula ) < oo and thus f is the sum of a

summable function fXa and a bounded function txa » . Now , since Tlo S1 , we

have \ A ...

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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