## Linear Operators, Part 1 |

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1x — v.26 22

that there exists a finite constant K such that for f in CBV , | ( Snf ) ( x ) SK ( v ( , [ 0 ,

27 ] ) + sup \ | ( x ) ] ) , 0 < x < 27 . OS XS2.12 23

1x — v.26 22

**Suppose**that ( Sn ) ( x ) → f ( x ) uniformly for every f in AC . Showthat there exists a finite constant K such that for f in CBV , | ( Snf ) ( x ) SK ( v ( , [ 0 ,

27 ] ) + sup \ | ( x ) ] ) , 0 < x < 27 . OS XS2.12 23

**Suppose**that ( i ) ( Sqf ) ( x ) = f ...Page 598

operator . Let à € 0 ( T ) , and lim , gn ( 2 ) + 0. Show that 2 is a pole of o ( T ) , and

that E ( 2 ; T ) X has a positive finite dimension . ( Hint . See Exercise VII.5.35 . ) ...

**Suppose**that gn ( T ) converges in the uniform operator topology to a compactoperator . Let à € 0 ( T ) , and lim , gn ( 2 ) + 0. Show that 2 is a pole of o ( T ) , and

that E ( 2 ; T ) X has a positive finite dimension . ( Hint . See Exercise VII.5.35 . ) ...

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surface of the unit sphere is SK . Putting h * ( x ) = supose < 1 h ( rx ) ) for ! 21 = 1 ,

show that there exists an absolute constant Cm , such that the integral of h * p ...

**Suppose**that for each r , 0 < r < l , the integral of the function \ h ( rx ) | ” over thesurface of the unit sphere is SK . Putting h * ( x ) = supose < 1 h ( rx ) ) for ! 21 = 1 ,

show that there exists an absolute constant Cm , such that the integral of h * p ...

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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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