## Linear Operators: General theory |

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Page 253

An equivalence class U of vectors wq will be said to

equivalence class V of vectors if there is a pair of ... Suppose that U and V are

up in the ...

An equivalence class U of vectors wq will be said to

**correspond**to anequivalence class V of vectors if there is a pair of ... Suppose that U and V are

**corresponding**equivalence classes and that u € U. Consider an arbitrary elementup in the ...

Page 254

It is thus seen that

same closed linear manifold M. Hence , if one of U and V is finite , M is finite

dimensional , and therefore the other of U and V is finite and has the same

cardinality .

It is thus seen that

**corresponding**equivalence classes U and V determine thesame closed linear manifold M. Hence , if one of U and V is finite , M is finite

dimensional , and therefore the other of U and V is finite and has the same

cardinality .

Page 281

Let kj , ... , k , be the indices

quasi - uniform convergence of { gk } . Then U ; = { s | \ 8x , ( 8 ) —fo ( s ) > { o } is

an open set containing s , for i = 1 , ... , r . Since A is dense in S , there exists a

point ...

Let kj , ... , k , be the indices

**corresponding**to £ o and k = 1 guaranteed by thequasi - uniform convergence of { gk } . Then U ; = { s | \ 8x , ( 8 ) —fo ( s ) > { o } is

an open set containing s , for i = 1 , ... , r . Since A is dense in S , there exists a

point ...

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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