## Linear Operators: General theory |

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Page 21

A set is said to be

nowhere

, if it contains a denumer- able

A set is said to be

**dense**in a topological space X, if its closure is X. It is said to benowhere

**dense**if its closure does not contain any open set. A space is separable, if it contains a denumer- able

**dense**set. 12 Theorem. // a topological space ...Page 451

We wish to prove that Z = n*Ği Z„ = n"_j Zni is

p, e/2), then SZ = <£. Hence U"=1 SZ^4 = 5. It follows from Theorem 1.6.9 ...

We wish to prove that Z = n*Ği Z„ = n"_j Zni is

**dense**in X. Suppose that Z is not**dense**in X, and that p 4 Z, Then some sphere 5(p, e) does not intersect Z. If S = 5(p, e/2), then SZ = <£. Hence U"=1 SZ^4 = 5. It follows from Theorem 1.6.9 ...

Page 842

1.7 (98) for measures, III.4.7 (128), III. 4.11 (130) Lebesgue decomposition, II 1.4.

14 (132) Saks decomposition, IV.9.7 (308) Yosida-Hewitt decomposition, (:J33)

De Morgan, rules of, (2)

...

1.7 (98) for measures, III.4.7 (128), III. 4.11 (130) Lebesgue decomposition, II 1.4.

14 (132) Saks decomposition, IV.9.7 (308) Yosida-Hewitt decomposition, (:J33)

De Morgan, rules of, (2)

**Dense**convex sets, V.7.27 (437)**Dense**linear manifolds,...

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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 closed linear manifold compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive Definition denote dense differential equations Doklady Akad element equivalent everywhere exists extended real valued extension fi(E finite dimensional finite number function f Hausdorff space Hence Hilbert space homeomorphism inequality integral interval Lebesgue measure Lemma linear functional linear map linear operator linear topological space LP(S measurable function 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 Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory topological space uniformly unique v(fi valued function Vber vector valued weakly compact zero