## Linear Operators: General theory |

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Page 193

12

(T, ET, X). Let E be a g-null set in R. Then for X-almost all t, the set E(t) = {s\[s, t] e

E) is a u-null set. Proof. By

12

**Lemma**. Let (R, ZR, g) be the □product of finite measure spaces (S, £, /u) and(T, ET, X). Let E be a g-null set in R. Then for X-almost all t, the set E(t) = {s\[s, t] e

E) is a u-null set. Proof. By

**Lemma**11 it may be assumed that the measure ...Page 699

2 Therefore -1/4* g-V t dx 2yji 71 Jo 2e~V V which proves (*) and completes the

proof of the

CPk. For technical reasons occurring later the following

...

2 Therefore -1/4* g-V t dx 2yji 71 Jo 2e~V V which proves (*) and completes the

proof of the

**lemma**. Q.E.D. We shall now state and prove the**lemma**referred to asCPk. For technical reasons occurring later the following

**lemma**is stated for what...

Page 700

<xmJo Jo » Jc Jo This shows that if the

; it is also true for any integer m g A;. Thus, to prove the

show that it is true if A: is even. If A: = 1 the

<xmJo Jo » Jc Jo This shows that if the

**lemma**is known to be true for an integer A; it is also true for any integer m g A;. Thus, to prove the

**lemma**, it will suffice toshow that it is true if A: is even. If A: = 1 the

**lemma**has already been proved ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

44 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

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 convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma linear map linear operator linear topological space LP(S measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space null set 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 Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero