## Linear Operators: General theory |

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Page 249

The

22 ...

The

**identity**x + ya + \ x - y2 = 2l212 + 2 \ y / ? , X , Y EH , called the parallelogram**identity**, follows immediately from the axioms . If 8 = inf la — k | the preceding**identity**shows that KEK ki - k ; ) 2 = 2 \ x - k ; 2 + 212 — k ; 12 — 4 \ x — ( ki + k ) /22 ...

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

and hence , by II . 1 . 18 , the set of x for which Tnxin → 0 is a closed linear

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

operator ...

The

**identity**Tп n - 1 = = A ( n ) – " - A ( n - 1 ) n shows that { Tn / n } is boundedand hence , by II . 1 . 18 , the set of x for which Tnxin → 0 is a closed linear

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

operator ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

31 other sections not shown

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algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero