Linear Operators: General theory |
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Page 51
Q.E.D. LEMMA . If a homomorphism of one topological group into another is
continuous anywhere , it is continuous . PROOF . Let the homomorphism f : G H
be continuous at x , and let y eG . If V is a neighborhood of f ( y ) , then , by
Lemma 2 ...
Q.E.D. LEMMA . If a homomorphism of one topological group into another is
continuous anywhere , it is continuous . PROOF . Let the homomorphism f : G H
be continuous at x , and let y eG . If V is a neighborhood of f ( y ) , then , by
Lemma 2 ...
Page 56
of the image of any neighborhood G of the element 0 in X contains a
neighborhood of the element 0 in Y . Since a - b is a continuous function of a and
b , there is a neighborhood M of 0 such that M – MCG . For every x € X , x / n → 0 ,
and so ...
of the image of any neighborhood G of the element 0 in X contains a
neighborhood of the element 0 in Y . Since a - b is a continuous function of a and
b , there is a neighborhood M of 0 such that M – MCG . For every x € X , x / n → 0 ,
and so ...
Page 572
Conversely , let f ( T ) = 0 ; then , by Theorem 11 , f ( o ( T ) ) = 0 . Let f be analytic
on a neighborhood U of o ( T ) . For each a co ( T ) , there is an e ( a ) > 0 such
that the sphere S ( a , e ( a ) ) CU . Since o ( T ) is compact , a finite set of spheres
S ...
Conversely , let f ( T ) = 0 ; then , by Theorem 11 , f ( o ( T ) ) = 0 . Let f be analytic
on a neighborhood U of o ( T ) . For each a co ( T ) , there is an e ( a ) > 0 such
that the sphere S ( a , e ( a ) ) CU . Since o ( T ) is compact , a finite set of spheres
S ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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Common terms and phrases
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 |