## Linear Operators: General theory |

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Page xvi

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

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

...

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

...

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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...

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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 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

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

5 other sections not shown

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero