## 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 be a complete metric space with metric q(x, y). Let {A„} 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 be a complete metric space with metric q(x, y). Let {A„} 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 func tion of a

and b, there is a neighborhood M of 0 such that M~J\I Q G For every x (S, xjn -*□

0, and so x e ...

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 func tion of a

and b, there is a neighborhood M of 0 such that M~J\I Q G For every x (S, xjn -*□

0, and so x e ...

Page 309

The set A = S — U JLj Et 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 JLj Et 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 Z with ...### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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a-field Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact operator complex numbers complex valued contains continuous functions continuous linear convex set Corollary countably additive Definition denote dense differential equations disjoint sets Doklady Akad Duke Math element equivalent everywhere exists extended real valued extension finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality interval Lebesgue measure lim sup linear functional 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 Proof properties proved real numbers Riesz Russian semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space Trans uniformly unique v(fi valued function Vber vector valued weakly compact