## Linear Operators: General theory |

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Page 21

A set is said to bejlense in a topological space X, if its closure is X. It is said to be

nowhere

, if it contains a denumer- able

A set is said to bejlense in a topological space X, if its closure is X. It is said to be

nowhere

**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 281

Let A be a

sequence {/„} of continuous functions converges at every point of A to a

continuous limit /„. Then {/„} converges to f0 at every point of S if and only if {/„}

and every ...

Let A be a

**dense**subset of a compact Hausdorff space S, and suppose that asequence {/„} of continuous functions converges at every point of A to a

continuous limit /„. Then {/„} converges to f0 at every point of S if and only if {/„}

and every ...

Page 842

(2331 De Morgan, rules of, (2)

manifolds. V .7.40-41 (438-439)

simple functions in Lr, 1 gp< oo, III.3.8 (125)

and L„.

(2331 De Morgan, rules of, (2)

**Dense**convex sets, V.7.27 (437)**Dense**linearmanifolds. V .7.40-41 (438-439)

**Dense**set, definition, 1.6.11 (21)**density**ofsimple functions in Lr, 1 gp< oo, III.3.8 (125)

**density**of continuous functions in TMand L„.

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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