## Linear Operators, Part 1 |

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Page 106

The functions totally

measurable on S are the functions in the closure TM ( S ) in F ( S ) of the u -

simple functions . If for every E in { with ulu , E ) < 00 , the product xet off with the

characteristic ...

The functions totally

**u**-**measurable**on S , or , if u is understood , totallymeasurable on S are the functions in the closure TM ( S ) in F ( S ) of the u -

simple functions . If for every E in { with ulu , E ) < 00 , the product xet off with the

characteristic ...

Page 119

( b ) the function g defined by g ( s ) = f ( s ) if s ¢S + US - , g ( s ) = 0 if se S + US - ,

is

extended real - valued ) which is defined only on the complement of a p - null set

NCS .

( b ) the function g defined by g ( s ) = f ( s ) if s ¢S + US - , g ( s ) = 0 if se S + US - ,

is

**u**-**measurable**. Next suppose that we consider a function † ( vector orextended real - valued ) which is defined only on the complement of a p - null set

NCS .

Page 178

XF ( s ) / ( s ) g ( s ) = lim xe , ( s ) / ( s ) g ( s ) , 8 6 S , and the pointwise limit of a

sequence of

Xx . fg is

...

XF ( s ) / ( s ) g ( s ) = lim xe , ( s ) / ( s ) g ( s ) , 8 6 S , and the pointwise limit of a

sequence of

**measurable**functions is**measurable**, it will suffice then to show thatXx . fg is

**measurable**. Thus we may and shall assume that v (**u**, F ) < 0 and v ( 2...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

25 other sections not shown

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### Common terms and phrases

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