## Linear Operators, Part 1 |

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Page 68

... y ) = 0 and x = y , by Corollary 14 . Q . E . D . 27 LEMMA . A weakly convergent

sequence { xm } of points in a normed linear space is bounded . Its limit x is in the

closed linear manifold

... y ) = 0 and x = y , by Corollary 14 . Q . E . D . 27 LEMMA . A weakly convergent

sequence { xm } of points in a normed linear space is bounded . Its limit x is in the

closed linear manifold

**determined**by { xn ) , and a < lim inf ( nl . n - 00 PROOF .Page 136

There is a uniquely

o - field containing a given family of sets . Proof . There is at least one field ,

namely the field of all subsets of S , which contains a given family t . The

intersection ...

There is a uniquely

**determined**smallest field and a uniquely**determined**smallesto - field containing a given family of sets . Proof . There is at least one field ,

namely the field of all subsets of S , which contains a given family t . The

intersection ...

Page 455

Let G = U21 Yn ; then f is

subset of X * , there exists a denumerable subset G , of X * such that each fe F is

can ...

Let G = U21 Yn ; then f is

**determined**by G . It follows that if F is a denumerablesubset of X * , there exists a denumerable subset G , of X * such that each fe F is

**determined**by GF . We can even assert that each continuous linear functional |can ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint 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 obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero