## Linear Operators: General theory |

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Page 286

1 discussion of the present section covers the space

,E,fi). 1 Theorem. // 1 < p < oo and l/p+l/? = 1. there is an isometric isomorphism

between L*(

1 discussion of the present section covers the space

**lp**as well as the space**LP**{**S**,E,fi). 1 Theorem. // 1 < p < oo and l/p+l/? = 1. there is an isometric isomorphism

between L*(

**S**, E, ft) and La(**S**, E, fi) in which corresponding vectors x* and g are ...Page 298

Since \U„\ ^ 1 it follows by III.8.8 that UJ -> / for all / in

, the theorem follows from Lemma IV.5.4. If /j.(S) < oo a similar argument proves

the desired result in LX(S, Z, fi). Q.E.D. If S is n-dimensional Euclidean space, ...

Since \U„\ ^ 1 it follows by III.8.8 that UJ -> / for all / in

**LP**(**S**, Z, fi). Hence if 1 p < oo, the theorem follows from Lemma IV.5.4. If /j.(S) < oo a similar argument proves

the desired result in LX(S, Z, fi). Q.E.D. If S is n-dimensional Euclidean space, ...

Page 336

Q.E.D. The natural ordering just used for functions in M(S, Z, fi) has already been

employed for functions in the spaces

8.23 we proved that if a subset of one of these spaces is bounded above by ...

Q.E.D. The natural ordering just used for functions in M(S, Z, fi) has already been

employed for functions in the spaces

**LP**(**S**, Z, fi), 1 ^ p ^ oo. In theorems 8.22 and8.23 we proved that if a subset of one of these spaces is bounded above by ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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a-finite Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear functional convex set Corollary countably additive Definition denote dense Doklady Akad element equivalent everywhere exists finite dimensional follows from Theorem function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lebesgue measure linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative non-zero normed linear space open set operator topology positive measure space Proc properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space valued function vector space weak topology weakly compact weakly sequentially compact zero