## Linear Operators: General theory |

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Page 244

Finite Dimensional Spaces The space En , as will be seen presently , is the

prototype of all n - dimensional normed linear spaces , and

observed first that En is a B - space . The Minkowski inequality ( III.3.3 ) shows En

to be ...

Finite Dimensional Spaces The space En , as will be seen presently , is the

prototype of all n - dimensional normed linear spaces , and

**hence**it should beobserved first that En is a B - space . The Minkowski inequality ( III.3.3 ) shows En

to be ...

Page 423

continuous at the origin , and

is weakly continuous , and y * € Y * . Then y * T is a linear functional on X which is

...

**Hence**y * ( Tr ) / < € , so that Tx e N ( 0 ; yi , ... , Y * , £ ) . Therefore , T is weaklycontinuous at the origin , and

**hence**at every point . Conversely , suppose that Tis weakly continuous , and y * € Y * . Then y * T is a linear functional on X which is

...

Page 441

Put K ; = co ( ( Titu ) nQ ) C9i + U . Then Ki is a closed , and

subset of co ( Q ) .

easy induction on Lemma 2.5 . It follows readily that p has the form p = < ! - 1 a ; k

,, a ...

Put K ; = co ( ( Titu ) nQ ) C9i + U . Then Ki is a closed , and

**hence**a compact ,subset of co ( Q ) .

**Hence**co ( Q ) = Co ( K , U ... UK ) = co ( KU ... UK , ) , by aneasy induction on Lemma 2.5 . It follows readily that p has the form p = < ! - 1 a ; k

,, a ...

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### Contents

Preliminary Concepts 1 1 Preliminary Concepts | 1 |

B Topological Preliminaries 10 B Topological Preliminaries | 10 |

Metric Spaces | 18 |

Copyright | |

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