## 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 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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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