## Linear Operators, Part 1 |

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of B , and B is said to contain A. This is denoted symbolically by A Ç B , or B 2 A.

Two sets are the same if and only if they have the same

and only if A C B and B ÇA . The set A is said to be a proper subset of the set B if

A ...

of B , and B is said to contain A. This is denoted symbolically by A Ç B , or B 2 A.

Two sets are the same if and only if they have the same

**elements**, i.e. , A = B ifand only if A C B and B ÇA . The set A is said to be a proper subset of the set B if

A ...

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The

satisfy the following conditions : ( i ) a ( be ) = ( ab ) , a , b , ce G ; ( ii ) there is an

a ...

The

**element**ab is called the product of a and b . The product ab is required tosatisfy the following conditions : ( i ) a ( be ) = ( ab ) , a , b , ce G ; ( ii ) there is an

**element**e in G , called the identity or the unit of G , such that ae = ea = a for everya ...

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A field is a commutative ring in which the non - zero

multiplication . The unit of this group in a field will be written as I instead of e . A

linear vector space , linear space , or vector space over a field Ø is an additive ...

A field is a commutative ring in which the non - zero

**elements**form a group undermultiplication . The unit of this group in a field will be written as I instead of e . A

linear vector space , linear space , or vector space over a field Ø is an additive ...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

### Common terms and phrases

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