## Linear Operators: General theory |

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Page 2

CA of B , and B is said to contain A . This is denoted symbolically by A CB , or B2

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

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

CA of B , and B is said to contain A . This is denoted symbolically by A CB , or B2

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

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

Page 34

The

satisfy the following conditions : ( i ) a ( bc ) = ( ab ) c , a , b , c e G ; ( ii ) there is an

...

The

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

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

Page 36

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

multiplication . The unit of this group in a field will be written as 1 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 1 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 |

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