## Linear Operators: General theory |

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Page 421

**Hence**n X e X. g ( x ) = { viti ( x ) , Q.E.D. PROOF OF THEOREM 9. Every functional in l ' is ' - continuous , by Lemma 8 . Conversely , let g + o be a linear functional on X which is I ' - continuous . There exists a I - neighborhood N ...Page 441

Put K ; = co ( ( qi + U ) NQ ) Çqi + U . Then K ; is a closed , and

Put K ; = co ( ( qi + U ) NQ ) Çqi + U . Then K ; is a closed , and

**hence**a compact , subset of co ( Q ) .**Hence**CO ( Q ) = co ( K , U ... UKN ) = co ( K , U ... UKn ) , by an easy induction on Lemma 2.5 . It follows readily that p has ...Page 485

**Hence**T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma 7 that T ** is continuous relative to the X * , Y *** topologies in X ** , Y ** , respectively . If S , S ** are the closed unit spheres in X ...### 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