## 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 {a„} in a

even' Cauchy sequence is convergent, a

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

A sequence {a„} in a

**metric space**is a Cauchy sequence if limm „ n(a„,, a„) = 0. Ifeven' Cauchy sequence is convergent, a

**metric space**is said to be complete. Thenext 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 | |

44 other sections not shown

### Other editions - View all

Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |

### Common terms and phrases

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 convex set Corollary countably additive Definition denote dense differential equations Doklady Akad Duke Math element equivalent exists finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral interval isometric isomorphism Lemma 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 properties proved real numbers reflexive Riesz Russian scalar semi-group sequentially compact Show simple functions subset subspace Suppose theory TM(S topological space Trans valued function Vber vector space weak topology weakly compact weakly sequentially compact zero