## Linear Operators, Part 1 |

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Page 259

Since a set in a complete metric space is compact if and only if it is closed and

purposes to state conditions for conditional compactness. The following

elementary ...

Since a set in a complete metric space is compact if and only if it is closed and

**conditionally compact**(I.6.15), it will be sufficient and convenient for our presentpurposes to state conditions for conditional compactness. The following

elementary ...

Page 266

Then K is

collection {E1, ..., E.} of sets with union S, and points s, in E, i = 1,..., m, such that

sup sup f(s)-f(s) < e, i = 1,..., n. feK se E, PRoof. If X is the field of all sets in S then

C(S) ...

Then K is

**conditionally compact**if and only if for every e > 0 there is a finitecollection {E1, ..., E.} of sets with union S, and points s, in E, i = 1,..., m, such that

sup sup f(s)-f(s) < e, i = 1,..., n. feK se E, PRoof. If X is the field of all sets in S then

C(S) ...

Page 267

Then K is

that sup f(s)—f(t) < e, 9(8, t) < 6. feK PRoof. If the condition is satisfied then K is an

equicontinuous family in C(S) and, by Theorem 7, K is compact. - Conversely ...

Then K is

**conditionally compact**if and only if for every e -> 0 there is a 6 × 0 suchthat sup f(s)—f(t) < e, 9(8, t) < 6. feK PRoof. If the condition is satisfied then K is an

equicontinuous family in C(S) and, by Theorem 7, K is compact. - Conversely ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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