## Linear Operators: General theory |

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Page 20

Thus A4 + X , and A is open , and

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

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

Thus A4 + X , and A is open , and

**contains**a sphere S , = S ( P1 , & j ) with 0 < & i< 1 / 2 . By assumption , the set A , 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 nM ...

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 nM ...

Page 309

The set A = S - U - E ; then

will now be shown that every measurable subset B of A

u ( F ) 3 € . Suppose , on the contrary , that some such set B

with ...

The set A = S - U - E ; 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 ) 3 € . Suppose , on the contrary , that some such set B

**contains**no set ofwith ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

21 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex Consequently constant contains converges convex Corollary defined DEFINITION denote dense determined differential 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 mean measure space metric neighborhood norm positive measure problem Proc projection PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement strongly subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero