## Linear Operators, Part 1 |

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Page 19

The

X , with its

such that there exists a

...

The

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

0. If every Cauchy sequence is convergent , a

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

A sequence { an } in a

**metric**space is a Cauchy sequence if limm , n olan , an ) =0. If every Cauchy sequence is convergent , a

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

Page 91

in admits an equivalent

invariant

complete linear

linear ...

in admits an equivalent

**metric**under which it is complete , then G admits aninvariant

**metric**and is complete under each invariant**metric**. Thus everycomplete linear

**metric**space can be metrized to be F - space . Further , a normedlinear ...

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

80 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

algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero