## Linear Operators: General theory |

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Page 20

Thus A , + X , and A is open , and

< 1 / 2 . By assumption , the set A2 does not

hence the non - void open set An S ( P1 , & 1 / 2 )

Thus A , + X , and A is open , and

**contains**a sphere S , = S ( P1 , & j ) with 0 < Ei< 1 / 2 . By assumption , the set A2 does not

**contain**the open set S ( P1 , & z / 2 ) ;hence the non - void open set An S ( P1 , & 1 / 2 )

**contains**a sphere S , = S ( P2 ...Page 56

of the image of any neighborhood G of the element 0 in X

neighborhood of the element 0 in Y . Since a - b is a continuous function of a and

b , there is a neighborhood M of 0 such that M – MCG . For every x e X , xin + 0 ,

and so X e ...

of the image of any neighborhood G of the element 0 in X

**contains**aneighborhood of the element 0 in Y . Since a - b is a continuous function of a and

b , there is a neighborhood M of 0 such that M – MCG . For every x e X , xin + 0 ,

and so X e ...

Page 309

The set A = S - UAE ; then

will now be shown that every measurable subset B of A

u ( F ) Sε . Suppose , on the contrary , that some such set B

...

The set A = S - UAE ; then

**contains**no atoms having measure greater than ε . Itwill now be shown that every measurable subset B of A

**contains**a set F with 0 <u ( F ) Sε . Suppose , on the contrary , that some such set B

**contains**no set of with...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assume 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 domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence 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