## Linear Operators, Part 1 |

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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

every Cauchy sequence is convergent, a

The next three lemmas are immediate consequences of the definitions. 6 LEMMA

.

A sequence {an} in a

**metric space**is a Cauchy sequence if limo., Q(a, , a,) = 0. Ifevery Cauchy sequence is convergent, a

**metric space**is said to be complete.The next three lemmas are immediate consequences of the definitions. 6 LEMMA

.

Page 21

Thus a subspace of a separable

be a base for the topology in a topological space X, and let p, e An. If V is open,

there is a base element A, contained in V, and hence a point p, in V. Thus the ...

Thus a subspace of a separable

**metric space**is separable. PRoof. Let A1, A2, ...be a base for the topology in a topological space X, and let p, e An. If V is open,

there is a base element A, contained in V, and hence a point p, in V. Thus the ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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