## Linear Operators: General theory |

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Page 19

The

X , with its

such that there exists a

...

The

**metric**topology in X is the smallest topology containing the spheres . The setX , with its

**metric**topology , is called a**metric**space . If X is a topological spacesuch that there exists a

**metric**function whose topology is the same as the original...

Page 20

A sequence { an } in a

0. If every Cauchy sequence is convergent , a

. The next three lemmas are immediate consequences of the definitions .

A sequence { an } in a

**metric**space is a Cauchy sequence if limm . n ( am , an ) =0. If every Cauchy sequence is convergent , a

**metric**space is said to be complete. The next three lemmas are immediate consequences of the definitions .

Page 91

admits an equivalent

invariant

complete linear

normed linear ...

admits an equivalent

**metric**under which it is complete , then G admits aninvariant

**metric**and is complete under each invariant**metric**. Thus everycomplete linear

**metric**space can be metrized to be an F - space . Further , anormed linear ...

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### Contents

B Topological Preliminaries 10 B Topological Preliminaries | 10 |

quences | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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