## 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, for each f in K, each t in N, and some i < n, f(s)-f(t) < f(s)-f(s)|+|f(s)-f(t)+f(t)-f(t)

& e. and so K is equicontinuous. Q.E.D. -> 8 CoRollARY. Let S be a compact

metric space and K be a bounded set in C(S). Then K is

Then, for each f in K, each t in N, and some i < n, f(s)-f(t) < f(s)-f(s)|+|f(s)-f(t)+f(t)-f(t)

& e. and so K is equicontinuous. Q.E.D. -> 8 CoRollARY. Let S be a compact

metric space and K be a bounded set in C(S). Then K is

**conditionally compact**if ...### What people are saying - Write a review

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

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