## Linear Operators: General theory |

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Page 28

If f is a generalized Cauchy sequence in a complete metric space X , there

a pe X such that lim f ( d ) = p . Proof . Let dn € D be such that c , ca 2 dn implies e

( / ( c ) , f ( cz ) ) < 1 / n . Let bn be an upper bound for the finite set { d , da , . . .

If f is a generalized Cauchy sequence in a complete metric space X , there

**exists**a pe X such that lim f ( d ) = p . Proof . Let dn € D be such that c , ca 2 dn implies e

( / ( c ) , f ( cz ) ) < 1 / n . Let bn be an upper bound for the finite set { d , da , . . .

Page 362

Under the hypotheses of Exercise 37 , show that there

341 ( x ) & n ( x ) dx if and only if the functions on - camnandu ( x ) , m 2 1 , are

uniformly bounded and equicontinuous . 39 Let { an } , - 00 < n < too , be a

bounded ...

Under the hypotheses of Exercise 37 , show that there

**exists**f in C with an = $341 ( x ) & n ( x ) dx if and only if the functions on - camnandu ( x ) , m 2 1 , are

uniformly bounded and equicontinuous . 39 Let { an } , - 00 < n < too , be a

bounded ...

Page 683

from Lemma 10 and Corollary 5.3 that the limit m = lim Mn

Corollary 5.2 , m ( q - le ) = m ( e ) , so that the map T : 10 ) → f ( $ ( . ) ) as an

operator in the space L ( S , E , m ) has its norm ( T1 = 1 ( Lemma 5.7 ) . Now let f

be a ...

from Lemma 10 and Corollary 5.3 that the limit m = lim Mn

**exists**in ca ( , u ) . ByCorollary 5.2 , m ( q - le ) = m ( e ) , so that the map T : 10 ) → f ( $ ( . ) ) as an

operator in the space L ( S , E , m ) has its norm ( T1 = 1 ( Lemma 5.7 ) . Now let f

be a ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

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