## 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

hence , by II . 1 . 18 , the set of x for which T ^ x / n + 0 is a closed linear manifold .

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

The

**identity**( * ) - п = A ( n ) - n - 1 - A ( n - 1 ) n shows that { T ] n } is bounded andhence , by II . 1 . 18 , the set of x for which T ^ x / n + 0 is a closed linear manifold .

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

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### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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