## Linear Operators: General theory |

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Page 54

A linear mapping of one F - space into another is continuous if and only if it maps

linear and continuous , and let B C X be

A linear mapping of one F - space into another is continuous if and only if it maps

**bounded**sets into**bounded**sets . Proof . Let X , Y be F - spaces , let T : x + y belinear and continuous , and let B C X be

**bounded**. For every neighborhood V of ...Page 97

and to be

function u is important because it dominates p in the sense that v ( u , E ) 2 u ( E )

for E € £ ; the reader should test his comprehension of Definition 4 below by ...

and to be

**bounded**if u is**bounded**. The total variation v ( u ) of an additive setfunction u is important because it dominates p in the sense that v ( u , E ) 2 u ( E )

for E € £ ; the reader should test his comprehension of Definition 4 below by ...

Page 345

if and only if it is

and each j = 0 , 1 , . . . , p . ( c ) Co is not weakly complete , and not reflexive . ( d )

A subset A CCP is conditionally compact if and only if it is

...

if and only if it is

**bounded**and f ( s ) converges ( to f ( ' \ ( s ) ) for each s in [ a , b ]and each j = 0 , 1 , . . . , p . ( c ) Co is not weakly complete , and not reflexive . ( d )

A subset A CCP is conditionally compact if and only if it is

**bounded**and for each...

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### Contents

Metric Spaces | 19 |

Convergence and Uniform Convergence of Generalized | 26 |

Exercises | 33 |

Copyright | |

10 other sections not shown

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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak zero