## Linear Operators: General theory |

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Page 19

The metric topology in X is the smallest topology containing the spheres . The set

X , with its metric topology , is called a

such that there exists a metric function whose topology is the same as the original

...

The metric topology in X is the smallest topology containing the spheres . The set

X , 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

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

.

A sequence { an } in a

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

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

.

Page 21

A space is separable , if it contains a denumerable dense set . 12 THEOREM . If a

topological space has a countable base , it is separable . Conversely , if a

A space is separable , if it contains a denumerable dense set . 12 THEOREM . If a

topological space has a countable base , it is separable . Conversely , if a

**metric****space**is separable , it has a countable base . Thus a subspace of a separable ...### What people are saying - Write a review

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### Contents

Special Spaces | 237 |

Convex Sets and Weak Topologies | 409 |

General Spectral Theory | 555 |

Copyright | |

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