## Linear Operators: General theory |

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Page 472

Šmulian ( 7 ] obtains two interesting necessary and

differentiability of the norm . THEOREM . In order that the norm is strongly

differentiable at a point æ in a B - space X , it is necessary and

every ...

Šmulian ( 7 ] obtains two interesting necessary and

**sufficient**conditions for strongdifferentiability of the norm . THEOREM . In order that the norm is strongly

differentiable at a point æ in a B - space X , it is necessary and

**sufficient**thatevery ...

Page 592

If 20 € D is such that T - 1 ( 20 ) exists , then it follows from Lemma 6 that for a

neighborhood of 1o . To prove the second part of the lemma we suppose that

there ...

If 20 € D is such that T - 1 ( 20 ) exists , then it follows from Lemma 6 that for a

**sufficiently**near 2o the inverse T - 1 ( 2 ) exists and is an analytic function of 2 in aneighborhood of 1o . To prove the second part of the lemma we suppose that

there ...

Page 637

By Lemma 20 ( c ) , for each w > W , there exists a constant Mo < such that y ( t ) =

\ PT ( t ) ) < Mewt for

integrable over every finite interval of the positive real axis . Thus , if we ...

By Lemma 20 ( c ) , for each w > W , there exists a constant Mo < such that y ( t ) =

\ PT ( t ) ) < Mewt for

**sufficiently**large t . On the other hand , we have seen that y isintegrable over every finite interval of the positive real axis . Thus , if we ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

81 other sections not shown

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### Common terms and phrases

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