## Linear Operators, Part 1 |

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Page 169

1

and a finite collection of disjoint sets An , ... , A , € & such that AqU ... U An E , v ( u

, E ) < € , and sup \ ( s ) -f ( t ) / < € , j = 1 , 8 , teA ; 2 Let A CS be a u - null set .

1

**Show**that fe TM ( S , E , u ) if and only if for each ε > 0 there exists a set E , €and a finite collection of disjoint sets An , ... , A , € & such that AqU ... U An E , v ( u

, E ) < € , and sup \ ( s ) -f ( t ) / < € , j = 1 , 8 , teA ; 2 Let A CS be a u - null set .

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projection S , in each of the spaces Lp , BV , CBV , AC , C ) , 13p 300 , k = 0 , 1 , 2

, . . . , 00.

**Show**that Snj is given by the formula ( S . } ) ( x ) = ** En ( x , y ) / ( y ) dy , and is aprojection S , in each of the spaces Lp , BV , CBV , AC , C ) , 13p 300 , k = 0 , 1 , 2

, . . . , 00.

**Show**that the range of Sn lies in C'16 ) . 3**Show**that S , “ I strongly in ...Page 360

21

c.o.n. system is localized if and only if max En ( x , y ) EM < oo for each a > 0 . 1x

— v.26 22 Suppose that ( Sn ) ( x ) → f ( x ) uniformly for every f in AC .

...

21

**Show**that if | S7 En ( x , x ) dz | M , then the convergence of Snf for a givenc.o.n. system is localized if and only if max En ( x , y ) EM < oo for each a > 0 . 1x

— v.26 22 Suppose that ( Sn ) ( x ) → f ( x ) uniformly for every f in AC .

**Show**that...

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### Contents

Preliminary Concepts | 1 |

The VitaliHahnSaks Theorem and Spaces of Measures | 7 |

B Topological Preliminaries | 10 |

Copyright | |

87 other sections not shown

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