Linear Operators, Part 1 |
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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 , glam , 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 , glam , 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
Preliminary Concepts | 1 |
The VitaliHahnSaks Theorem and Spaces of Measures | 7 |
B Topological Preliminaries | 10 |
Copyright | |
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Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero
Bibliographic information
| Title | Linear Operators, Part 1 Linear Operators, Jacob T. Schwartz Volume 7 of Mathematics, Pure, Applied Volume 7 of Pure and applied mathematics, ISSN 0079-8185 Volume 7 of Pure and applied mathematics. A series of texts and monographs |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of California |
| Digitized | 26 Jun 2018 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |