## Linear Operators, Part 1 |

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Page 188

Q.E.D. As in the case of finite measure spaces we shall call the measure space (

S , E , u ) constructed in

) the product measure space and write ( S , E , u ) = Ph_ ( Si , Ei , Mi ) . The best ...

Q.E.D. As in the case of finite measure spaces we shall call the measure space (

S , E , u ) constructed in

**Corollary**6 from the o - finite measure spaces ( Si , Einpi) the product measure space and write ( S , E , u ) = Ph_ ( Si , Ei , Mi ) . The best ...

Page 246

8

PROOF . In the notation of

and thus for *** in X ** we have x ** v * = x * x , where x = { n - 1 *** { = 1 *** ( 0,1 )

...

8

**COROLLARY**. A finite dimensional normed linear space is reflexive . 4 1PROOF . In the notation of

**Corollary**7 , n n x * x 25 * ( x ) ** ( b ; ) , x * 26 ** ( bi ) ,and thus for *** in X ** we have x ** v * = x * x , where x = { n - 1 *** { = 1 *** ( 0,1 )

...

Page 422

11

total subspace of X * . Then the following statements are equivalent : ( i ) f is in l ' ;

( ii ) f is I - continuous ; ( iii ) { x \ / ( x ) = 0 } is I ' - closed . Proof . By Theorem 9 ...

11

**COROLLARY**. Let f be a linear functional on the linear space X , and let be atotal subspace of X * . Then the following statements are equivalent : ( i ) f is in l ' ;

( ii ) f is I - continuous ; ( iii ) { x \ / ( x ) = 0 } is I ' - closed . Proof . By Theorem 9 ...

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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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