## Linear Operators: General theory |

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Page 39

... if F is a totally ordered subsystem of E , then it is easy to see that UF is a right

ideal in R which does not contain e . Zorn ' s lemma ( 2 . 7 ) applies to give a

maximal element in E , which is readily

containing lo .

... if F is a totally ordered subsystem of E , then it is easy to see that UF is a right

ideal in R which does not contain e . Zorn ' s lemma ( 2 . 7 ) applies to give a

maximal element in E , which is readily

**seen**to be a maximal right idealcontaining lo .

Page 254

Now since uge U there is a finite chain of the form given in [ * ] above in which

successive vectors have non - zero scalar products . Thus by forming the chain

VR , U « » . . . , Ume ' s Up . it is

is ...

Now since uge U there is a finite chain of the form given in [ * ] above in which

successive vectors have non - zero scalar products . Thus by forming the chain

VR , U « » . . . , Ume ' s Up . it is

**seen**that vp is equivalent to Vg , and thus that Vgis ...

Page 690

... it is

Pi = 1 , Pl . 31 , and T " ( t , : ) < Pn ( ( o ) ) , ) , n = 1 , 2 , . . . . By the Riesz convexity

theorem we have also P , 31 , and , since Pn = Em Pm – - 1 pm , it is

... it is

**seen**that g is in Ly . By Lemma 6 . 4 there is a positive operator Pin Ly withPi = 1 , Pl . 31 , and T " ( t , : ) < Pn ( ( o ) ) , ) , n = 1 , 2 , . . . . By the Riesz convexity

theorem we have also P , 31 , and , since Pn = Em Pm – - 1 pm , it is

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### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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