## Linear Operators, Part 1 |

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Page 60

Statement ( iv ) clearly

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

0 , 12 51 Ta ] = [ 01 | 7 ) > M [ 2 ] . 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 51 Ta ] = [ 01 | 7 ) > M [ 2 ] . 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

To each point pe C there corresponds a unique nearest point N ( P ) € K . To see

this , note that Lemma IV . 4 . 2

limina p - k ; [ = inf ker \ p - k | then { k ; } converges , say , to qeK . If { k { } is

another ...

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 { ki } is a sequence in K such thatlimina p - k ; [ = inf ker \ p - k | then { k ; } converges , say , to qeK . If { k { } is

another ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm obtained operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero