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 metric space . If X is a topological space
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 space
such that there exists a metric function whose topology is the same as the original
...
Page 20
A sequence { an } in a metric space is a Cauchy sequence if limm . n g ( am , 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
.
A sequence { an } in a metric space is a Cauchy sequence if limm . n g ( am , 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
admits an equivalent metric under which it is complete , then G admits an
invariant metric and is complete under each invariant metric . Thus every
complete linear metric space can be metrized to be an F - space . Further , a
normed linear ...
admits an equivalent metric under which it is complete , then G admits an
invariant metric and is complete under each invariant metric . Thus every
complete linear metric space can be metrized to be an F - space . Further , a
normed linear ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm operator positive measure problem Proc proof properties proved respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topological space topology transformations u-measurable uniform uniformly unique unit valued vector weak weakly compact zero
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Bibliographic information
| Title | Linear Operators: General theory Part 1 of Linear Operators, Jacob T. Schwartz Volume 1; Volume 7 of Pure and applied mathematics : a series of texts and monographs Volume 7 of Pure and applied mathematics Interscience Press Pure and applied mathematics, ISSN 0079-8185 Wiley classics library |
| Authors | Nelson Dunford, Jacob T. Schwartz, William G. Bade |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of Michigan |
| Digitized | 20 Nov 2007 |
| ISBN | 0470226056, 9780470226056 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |