## Linear Operators: General theory |

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Page 20

(The Baire category theorem) If a complete metric space is the denumerable

union of closed subsets, at least one of these closed subsets

open set. Proof. Let X he a complete metric space with metric q(x, y). Let {An} be

a ...

(The Baire category theorem) If a complete metric space is the denumerable

union of closed subsets, at least one of these closed subsets

**contains**a non-voidopen set. Proof. Let X he a complete metric space with metric q(x, y). Let {An} be

a ...

Page 56

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

neighborhood of the element 0 in ?)• Since a— b is a continuous function of a

and b, there is a neighborhood M of 0 such that M—MQG. For every x e 36, xjn -*

□ 0, and so if ...

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

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

and b, there is a neighborhood M of 0 such that M—MQG. For every x e 36, xjn -*

□ 0, and so if ...

Page 309

The set A = S — U J=1 F, then

It will now be shown that every measurable subset B of A

fi(F) :£ e. Suppose, on the contrary, that some such set B

The set A = S — U J=1 F, then

**contains**no atoms having measure greater than e.It will 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 ...### 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