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 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
.
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 21
A space is separable , if it contains a denumerable dense set . 12 THEOREM . If 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 separable ...
A space is separable , if it contains a denumerable dense set . 12 THEOREM . If 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 separable ...
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Contents
Metric Spaces | 19 |
Convergence and Uniform Convergence of Generalized | 26 |
Exercises | 33 |
Copyright | |
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Acad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint Doklady Akad domain elements 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 mean measure space metric space neighborhood norm operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology transformations u-integrable u-measurable uniformly union unique unit valued vector weak 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 |