## Linear Operators, Part 1 |

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Page 165

If f is a ul-integrable simple function, it is evident that f is also a u-integrable

simple function, and that so f(s)u,(ds)=se ... It follows from (d) and (e) that a

function f e L,(S, 21, ui, 3) is also in

is the ...

If f is a ul-integrable simple function, it is evident that f is also a u-integrable

simple function, and that so f(s)u,(ds)=se ... It follows from (d) and (e) that a

function f e L,(S, 21, ui, 3) is also in

**Lp**(**S**, 2, u, *), and that its norm in both spacesis the ...

Page 302

We say that fe

If f, and f2 are two real or complex valued functions in

0 we write fi > f or f. s fi. This clearly is a partial ordering (I.2.1) in L,. We now ...

We say that fe

**Lp**(**S**, 2, u) is positive and write f = 0 if f(s) > 0 for u-almost all s e S.If f, and f2 are two real or complex valued functions in

**Lp**(**S**, 2, u) such that fi-fo =0 we write fi > f or f. s fi. This clearly is a partial ordering (I.2.1) in L,. We now ...

Page 336

We shall first show that the lattice L of all f in M(S, X, u) with 0 < f < g has the

property that every well-ordered subset is countable. ... Furthermore any set B

which is bounded in the lattice

same ...

We shall first show that the lattice L of all f in M(S, X, u) with 0 < f < g has the

property that every well-ordered subset is countable. ... Furthermore any set B

which is bounded in the lattice

**Lp**(**S**, 2 u) contains a countable subset having thesame ...

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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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additive set function algebra Amer analytic arbitrary B-space Banach baſs Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complete complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense differential Doklady Akad element equation equivalent exists finite dimensional function defined function f g-field g-finite Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure Lemma Let f lim ſº linear map linear operator linear topological space Lp(S Math measurable functions measure space metric space Nauk SSSR N.S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc properties proved real numbers Riesz scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact zero