## Linear Operators: General theory |

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Statement ( iv ) clearly

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 , 13131 Tx - 1x T 7 ( 3 ) < Mel .Page 280

That ( 1 )

That ( 1 )

**implies**( 2 ) can be proved in a manner similar to that used in Theorem 14 to show that condition ( 3 ) of that theorem**implies**( 4 ) . From Corollary 19 it follows that S may be embedded as a dense subset of a compact ...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

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 lim ; + p - kil = inf zer \ p - k | then { k ; } converges , say , to qe ...### What people are saying - Write a review

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### Contents

B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc PROOF properties proved respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero