## Linear Operators, Part 1 |

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Page 114

integrable simple functions ; to prove it in the general case , let In be a sequence

of finitely valued u - simple functions which determine g . Then the functions Tin (

) are ...

**Statement**( b ) is clear from the definitions .**Statement**( c ) is evident for u -integrable simple functions ; to prove it in the general case , let In be a sequence

of finitely valued u - simple functions which determine g . Then the functions Tin (

) are ...

Page 447

By 1.8 ( c ) and 1.8 ( e ) , we have 1 { f ( x + ay ) – f ( x ) } < k ( y ) , a which implies (

a ) .

x + ayz ) . 2

By 1.8 ( c ) and 1.8 ( e ) , we have 1 { f ( x + ay ) – f ( x ) } < k ( y ) , a which implies (

a ) .

**Statement**( b ) follows from the inequality 2f ( x + ( yı + y2 ) ) = f ( x + ayı ) + F (x + ayz ) . 2

**Statement**( c ) is trivial .**Statement**( d ) follows from the inequality ...Page 487

in S. Hence , by IV.6.6 , the condition is equivalent to the

an equicontinuous subset of C ( S * ) . It follows from Theorem IV.6.7 , that T ( S )

is conditionally compact in the metric of Y if and only if the condition is satisfied .

in S. Hence , by IV.6.6 , the condition is equivalent to the

**statement**that T ( S ) isan equicontinuous subset of C ( S * ) . It follows from Theorem IV.6.7 , that T ( S )

is conditionally compact in the metric of Y if and only if the condition is satisfied .

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

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