## Linear Operators: General theory |

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Page 36

A linear

group X together with an operation m : 0 X X X , written as m ( Q , x ) = ax , which

satisfy the following four conditions : ( i ) alæ + y ) = ax + ay , AEO , X , y eX ; ( ii ) ...

A linear

**vector**space , linear space , or**vector**space over a field Ø is an additivegroup X together with an operation m : 0 X X X , written as m ( Q , x ) = ax , which

satisfy the following four conditions : ( i ) alæ + y ) = ax + ay , AEO , X , y eX ; ( ii ) ...

Page 250

For an arbitrary

) = y * * ( y1 , y ) / y * yı = y * x , which proves the existence of the desired y . To

see that y is unique , let y ' be an element of H such that y * x = ( x , y ' ) for every x

...

For an arbitrary

**vector**x in H the**vector**x — ( y * x ) / ( y * yılyı is in M so that ( x , y) = y * * ( y1 , y ) / y * yı = y * x , which proves the existence of the desired y . To

see that y is unique , let y ' be an element of H such that y * x = ( x , y ' ) for every x

...

Page 795

On the one - dimensional translation group and semi - group in certain function

spaces . Dissertation , University of Uppsala ( 1950 ) . Math . Rev . 12 , 108 (

1951 ) . Ogasawara , T . 1 . Compact metric Boolean algebras and

On the one - dimensional translation group and semi - group in certain function

spaces . Dissertation , University of Uppsala ( 1950 ) . Math . Rev . 12 , 108 (

1951 ) . Ogasawara , T . 1 . Compact metric Boolean algebras and

**vector**lattices .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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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