## Linear Operators, Part 1 |

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Page 262

Since the integral [ st ( s ) u ( ds ) satisfies the inequality Ist ( s ) u ( ds ) S sup | | ( 8

) [ 0 ( 41 , S ) , it is

) . The following theorem is a converse to this statement . 2 THEOREM .

Since the integral [ st ( s ) u ( ds ) satisfies the inequality Ist ( s ) u ( ds ) S sup | | ( 8

) [ 0 ( 41 , S ) , it is

**clear**that the integral is a continuous linear functional on C ( S) . The following theorem is a converse to this statement . 2 THEOREM .

Page 282

It is

function f is said to be almost periodic if it is continuous and if for every e > 0 there

is an L = L ( 8 ) > 0 such that every interval in R of length L contains at least one ...

It is

**clear**that T ( 8 ) CT ( S ) if ε < 8 and that - t € T ( 8 ) whenever te T ( € ) . Thefunction f is said to be almost periodic if it is continuous and if for every e > 0 there

is an L = L ( 8 ) > 0 such that every interval in R of length L contains at least one ...

Page 292

It is also

0 , then Fi UF , € Eg . It follows that Ez is a field . If { Fx } is a sequence of disjoint

elements of Eg with union F and if E4 € E , and E c E2 , then , by hypothesis , lim

...

It is also

**clear**that if F , ε Σ3 , then S - F , € E3 , and that if F1 , F , E E , with F F , =0 , then Fi UF , € Eg . It follows that Ez is a field . If { Fx } is a sequence of disjoint

elements of Eg with union F and if E4 € E , and E c E2 , then , by hypothesis , lim

...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

12 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 |

### Common terms and phrases

Akad 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 isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric 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