## Linear Operators: General theory |

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Page 19

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

X, with its metric topology, is called a

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

...

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

X, with its metric topology, is called a

**metric space**. If X is a topological spacesuch that 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 p(am, «„) = 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 | 34 |

Copyright | |

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### Common terms and phrases

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