## Linear Operators: General theory |

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Page 60

2 x E This shows that ( i )

2 x E This shows that ( i )

**implies**( iv ) . 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 ...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 bek P - k then { k ; } converges , say , to qe K. If ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain 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 mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero