## Linear Operators: General theory |

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Page 20

Thus Ax # X, and A\ is open, and

assumption, the set At does not

void open set A2 fl S(pv SJ2)

Thus Ax # X, and A\ is open, and

**contains**a sphere Sl = S(pv with 0 < e1 < 1/2. Byassumption, the set At does not

**contain**the open set S(pv ej'2); hence the non-void open set A2 fl S(pv SJ2)

**contains**a sphere S2 = S(p2, e2) with 0 < Łg < 1/2*.Page 56

of the image of any neighborhood G of the element 0 in X

neighborhood of the element 0 in 9). Since a— b is a continuous function of a

and b, there is a neighborhood M of 0 such that M — M Q G. For every x e S, x/n -

>• 0, and so ...

of the image of any neighborhood G of the element 0 in X

**contains**aneighborhood of the element 0 in 9). Since a— b is a continuous function of a

and b, there is a neighborhood M of 0 such that M — M Q G. For every x e S, x/n -

>• 0, and so ...

Page 309

The set A = S— U J_i Et then

will now be shown that every measurable subset B of A

{F) e. Suppose, on the contrary, that some such set B

The set A = S— U J_i Et then

**contains**no atoms having measure greater than e. Itwill now be shown that every measurable subset B of A

**contains**a set F with 0 < fi{F) e. Suppose, on the contrary, that some such set B

**contains**no set of E with ...### What people are saying - Write a review

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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