## Linear Operators: General theory |

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The

The

**identity**\ x + ya + lx -— y2 = 2 [ 2 +2 \ ylo , called the parallelogram**identity**, follows immediately from the axioms . If 8 = inflx - k | the preceding**identity**shows that X , Y EH , kek 3 ki - k ; \ 2 = 2 ( x - k ; / 2 + 2x – k ...Page 479

When these inverses exist , ( T - 1 ) * = ( T * ) - 1 PROOF . If T - l 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 - l exists and is in B ( Y , X ) , then , by Lemma 4 , ( TT - 1 ) * = ( T - 1 ) * T * is the

**identity**in Y * and ( T - IT ) * = T * ( T - 1 ) * is the**identity**on ...Page 661

The

The

**identity**Tn ( * ) n - 1 An ) - An - 1 ) n n shows that { T " / n } is 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 ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

quences | 26 |

Copyright | |

80 other sections not shown

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### Common terms and phrases

Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain 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 mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero