## Linear Operators: General theory |

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Statement ( iv ) clearly

i ) , ( ii ) , and ( iv ) are equivalent . If M = sup ( Tx ] is finite , then for an arbitrary x

+ 0 , 12 31 \ Tx ] = v1 TG ) SMļa . This shows that ( iii )

Statement ( iv ) clearly

**implies**the continuity of T at 0 ; so ( iv )**implies**( ii ) . This (i ) , ( ii ) , and ( iv ) are equivalent . If M = sup ( Tx ] is finite , then for an arbitrary x

+ 0 , 12 31 \ Tx ] = v1 TG ) SMļa . This shows that ( iii )

**implies**( iv ) . It is obvious ...Page 280

That ( 1 )

14 to show that condition ( 3 ) of that theorem

follows that S may be embedded as a dense subset of a compact Hausdorff

space ...

That ( 1 )

**implies**( 2 ) can be proved in a manner similar to that used in Theorem14 to show that condition ( 3 ) of that theorem

**implies**( 4 ) . From Corollary 19 itfollows that S may be embedded as a dense subset of a compact Hausdorff

space ...

Page 454

Nelson Dunford, Jacob T. Schwartz. Proof . To each point pe C there corresponds

a unique nearest point N ( p ) € K . To see this , note that Lemma IV . 4 . 2

that if { k ; } is a sequence in K such that limit p - k ; ' = inf zer \ p - k then { k ...

Nelson Dunford, Jacob T. Schwartz. Proof . To each point pe C there corresponds

a unique nearest point N ( p ) € K . To see this , note that Lemma IV . 4 . 2

**implies**that if { k ; } is a sequence in K such that limit p - k ; ' = inf zer \ p - k then { k ...

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### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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