## Linear Operators: General theory |

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Thus S ( x , 2 - K ) CUm , which shows that the

Thus S ( x , 2 - K ) CUm , which shows that the

**identity**mapping of X with the metric topology onto X with the given topology is continuous . Hence the**identity**is a homeomorphism and the space is metrizable . Q.E.D. W - y 7.Page 479

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

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 ( T - 1T ) * = T * ( T - 1 ) ...Page 661

The

The

**identity**' - 1 Tn n - 1 ( * ) A ( n ) A ( n - 1 ) n n shows that { T " / n } is bounded and hence , by II.1.18 , the set of x for which Trę / n +0 is a closed linear manifold . Thus X , is a closed linear manifold and , since a ...### What people are saying - Write a review

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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