## Linear Operators: General theory |

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Page 26

Hence the

- * 7 . Convergence and Uniform Convergence of Generalized Sequences The

notion of convergence introduced in Definition 6 . 5 is not sufficiently general for ...

Hence the

**identity**is a homeomorphism and the space is metrizable . Q . E . D . w- * 7 . Convergence and Uniform Convergence of Generalized Sequences The

notion of convergence introduced in Definition 6 . 5 is not sufficiently general for ...

Page 479

... if and only if its adjoint T * has a bounded inverse ( T * ) - 1 defined on all of X *

. When these inverses exist , ( T - 1 ) * = ( T * ) - 1 PROOF . If T - 1 exists and is in

B ( Y , X ) , then , by Lemma 4 , ( TT - 1 ) * = ( T - 1 ) * T * is the

... if and only if its adjoint T * has a bounded inverse ( T * ) - 1 defined on all of X *

. When these inverses exist , ( T - 1 ) * = ( T * ) - 1 PROOF . If T - 1 exists and is in

B ( Y , X ) , then , by Lemma 4 , ( TT - 1 ) * = ( T - 1 ) * T * is the

**identity**in Y * and ...Page 661

The

bounded and hence , by II . 1 . 18 , the set of x for which Tnx / n → 0 is a closed

linear manifold . Thus X , is a closed linear manifold and , since a continuous

linear ...

The

**identity**T ' n ( * ) - = A ( n ) - n - 1 - n A ( n - 1 ) n shows that { T " \ n } isbounded and hence , by II . 1 . 18 , the set of x for which Tnx / n → 0 is a closed

linear manifold . Thus X , is a closed linear manifold and , since a continuous

linear ...

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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