## Linear Operators, Part 1 |

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Page 421

By II.3.11 , it can be extended to a linear functional y , on En . By IV.3.7 , y , has

the form The map y n [ yı , ... , yn ] Qiyi i = 1

X. Q.E.D. + PROOF OF THEOREM 9. Every functional in I ' is I - continuous , by ...

By II.3.11 , it can be extended to a linear functional y , on En . By IV.3.7 , y , has

the form The map y n [ yı , ... , yn ] Qiyi i = 1

**Hence**n g ( x ) = ant . ( x ) , x ) X eX. Q.E.D. + PROOF OF THEOREM 9. Every functional in I ' is I - continuous , by ...

Page 441

Put co ( ( qi + UnQ ) C9i + U . Then K ; is a closed , and

of co ( Q ) .

induction on Lemma 2.5 . It follows readily that р has the { = 1 a ; kı , a ; 20 , 21-10

...

Put co ( ( qi + UnQ ) C9i + U . Then K ; is a closed , and

**hence**a compact , subsetof co ( Q ) .

**Hence**CO ( Q ) = co ( K , U ... UK ) = co ( KU ... UK ) . by an easyinduction on Lemma 2.5 . It follows readily that р has the { = 1 a ; kı , a ; 20 , 21-10

...

Page 485

compact if and only if its adjoint is weakly compact . Proof . Let T be weakly

compact . Since the closed unit sphere S * of Y * is y - compact ( V.4.2 ) , it follows

from ...

**Hence**Yni ? 8 THEOREM . ( Gantmacher ) An operator in B ( x , y ) is weaklycompact if and only if its adjoint is weakly compact . Proof . Let T be weakly

compact . Since the closed unit sphere S * of Y * is y - compact ( V.4.2 ) , it follows

from ...

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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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