## Linear Operators: General theory |

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Page 19

The metric topology in is the smallest topology containing the spheres. The set X,

with its metric topology, is called a

that there exists a metric function whose topology is the same as the original ...

The metric topology in is the smallest topology containing the spheres. The set X,

with its metric topology, is called a

**metric space**. If X is a topological space suchthat there exists a metric function whose topology is the same as the original ...

Page 20

A sequence {an} in a

every Cauchy sequence is convergent, a

The next three lemmas are immediate consequences of the definitions. 6 Lemma.

A sequence {an} in a

**metric space**is a Cauchy sequence if limm ,( Q(a„,, a„) = 0. Ifevery Cauchy sequence is convergent, a

**metric space**is said to be complete.The next three lemmas are immediate consequences of the definitions. 6 Lemma.

Page 21

A space is separable, if it contains a denumer- able dense set. 12 Theorem. // a

topological space has a countable base, it is separable. Conversely, if a

metric ...

A space is separable, if it contains a denumer- able dense set. 12 Theorem. // a

topological space has a countable base, it is separable. Conversely, if a

**metric****space**is separable, it has a countable base. Thus a subspace of a separablemetric ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries 84 | 34 |

Copyright | |

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### Common terms and phrases

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