## Linear Operators: General theory |

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Page 3

that is , for every a € A , the

g : B → C , then the mapping gf : A → C is defined by the equation ( gf ) ( a ) = g ( /

( a ) ) for a € A . If f : A → B and CCA , the symbol f ( C ) is used for the set of all ...

that is , for every a € A , the

**function f**assigns an element f ( a ) € B . If f : A + B andg : B → C , then the mapping gf : A → C is defined by the equation ( gf ) ( a ) = g ( /

( a ) ) for a € A . If f : A → B and CCA , the symbol f ( C ) is used for the set of all ...

Page 104

11 ) the addition and scalar multiplication of equivalence classes is well defined .

It is customary to speak of the elements

functions rather than sets of equivalent functions and this we shall ordinarily do .

11 ) the addition and scalar multiplication of equivalence classes is well defined .

It is customary to speak of the elements

**of F**( S , E , M , X ) as if they werefunctions rather than sets of equivalent functions and this we shall ordinarily do .

Page 196

Next we study the relation between the theory

theory

and

Next we study the relation between the theory

**of**product measures and thetheory

**of**vector valued integrals . ... Suppose that ( S , E , u ) is a measure spaceand

**F**is a u - measurable**function**whose values are in L ( T , ET , 2 ) , 1 sp < 00 .### What people are saying - Write a review

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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