## Linear Operators: General theory |

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Page 40

If Ø is a field , then a set X is said to be an

a vector space over Ø and if a ( xy ) = ( axby = x ( ay ) , α , με Χ , αε Φ . A right ( left

, two - sided ) ideal in an

If Ø is a field , then a set X is said to be an

**algebra**over Ø if X is a ring as well asa vector space over Ø and if a ( xy ) = ( axby = x ( ay ) , α , με Χ , αε Φ . A right ( left

, two - sided ) ideal in an

**algebra**is a right ( left , two - sided ) ideal in the ring ...Page 44

Thus the concepts of Boolean

If B and C are Boolean algebras and h : B → C , then h is said to be a

homomorphism , or a Boolean

, h ( x vy ) ...

Thus the concepts of Boolean

**algebra**and Boolean ring with unit are equivalent .If B and C are Boolean algebras and h : B → C , then h is said to be a

homomorphism , or a Boolean

**algebra**homomorphism , if h ( x ^ y ) = h ( x ) h ( y ), h ( x vy ) ...

Page 274

Let S be a compact Hausdorff space and C ( S ) be the

continuous functions on S. Let A be a closed subalgebra of C ( S ) which contains

the unit e and contains , with f , its complex conjugate f defined by F ( 8 ) = f ( s ) .

Let S be a compact Hausdorff space and C ( S ) be the

**algebra**of all complexcontinuous functions on S. Let A be a closed subalgebra of C ( S ) which contains

the unit e and contains , with f , its complex conjugate f defined by F ( 8 ) = f ( s ) .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element 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 o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen 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