## Linear Operators, Part 1 |

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Page 147

Q.E.D. 8 LEMMA. If (S. 2, u) is a measure space, a function f defined on S to 3 is a

null function if and only if it

f(s)u (ds) = 0 for every E in X if and only if f

Q.E.D. 8 LEMMA. If (S. 2, u) is a measure space, a function f defined on S to 3 is a

null function if and only if it

**vanishes**almost everywhere. If f is u-integrable then J.f(s)u (ds) = 0 for every E in X if and only if f

**vanishes**almost everywhere.Page 572

Thus, if U, is the union of those spheres S(x1, e(x,)) which contain an infinite

number of points of g(T), then f

but a finite number of isolated points of a(T), which we suppose to be the points {

A1, ...

Thus, if U, is the union of those spheres S(x1, e(x,)) which contain an infinite

number of points of g(T), then f

**vanishes**identically on U1. Hence, U1 contains allbut a finite number of isolated points of a(T), which we suppose to be the points {

A1, ...

Page 593

However, it is not obvious that the convergence of {f,(T)} can be deduced if f

possibility. 1 THEOREM. Let f, f, be in 7 (T), and let {f(T)f,(T)} converge to zero in

the uniform ...

However, it is not obvious that the convergence of {f,(T)} can be deduced if f

**vanishes**at certain points of g(T). The next theorem is concerned with thispossibility. 1 THEOREM. Let f, f, be in 7 (T), and let {f(T)f,(T)} converge to zero in

the uniform ...

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### Contents

A Settheoretic Preliminaries | 1 |

Convergence and Uniform Convergence of Generalized | 26 |

Algebraic Preliminaries | 34 |

Copyright | |

27 other sections not shown

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