## Linear Operators: General theory |

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Page 263

Since G is an

F1 -- G ) . If F is a closed set it follows from this inequality , by allowing G , to

range over all open sets containing FF1 , that Mz ( F ) SM ( FF ) + u2 ( F1 - F ) .

Since G is an

**arbitrary**open set containing F1 - G we have Mi ( Fi ) S2 ( G ) + M7 (F1 -- G ) . If F is a closed set it follows from this inequality , by allowing G , to

range over all open sets containing FF1 , that Mz ( F ) SM ( FF ) + u2 ( F1 - F ) .

Page 281

Let S be an

and only if it is bounded and , together with every subsequence , converges to to

quasi - uniformly on S . PROOF . The sequence { fn } in B ( S ) converges weakly

to ...

Let S be an

**arbitrary**set . A sequence { { n } in B ( S ) converges weakly to to ifand only if it is bounded and , together with every subsequence , converges to to

quasi - uniformly on S . PROOF . The sequence { fn } in B ( S ) converges weakly

to ...

Page 476

... Y ) , [ ( T – R ) x1 < E , X e A } where A is an

O is

converges to T if and only if { T _ x } converges to Tx for every æ in X . 3

DEFINITION .

... Y ) , [ ( T – R ) x1 < E , X e A } where A is an

**arbitrary**finite subset of X , and ε >O is

**arbitrary**. Thus , in the strong topology , a generalized sequence { Ta }converges to T if and only if { T _ x } converges to Tx for every æ in X . 3

DEFINITION .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

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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 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