Linear Operators: General theory |
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Page 60
Statement ( iv ) clearly implies the continuity of T at 0 ; so ( iv ) implies ( ii ) . This (
i ) , ( ii ) , and ( iv ) are equivalent . If M = sup ( Tx ] is finite , then for an arbitrary x
= 0 , la si Tx ] = [ v TC . ) SM \ x } . This shows that ( iii ) implies ( iv ) . It is obvious ...
Statement ( iv ) clearly implies the continuity of T at 0 ; so ( iv ) implies ( ii ) . This (
i ) , ( ii ) , and ( iv ) are equivalent . If M = sup ( Tx ] is finite , then for an arbitrary x
= 0 , la si Tx ] = [ v TC . ) SM \ x } . This shows that ( iii ) implies ( iv ) . It is obvious ...
Page 280
That ( 1 ) implies ( 2 ) can be proved in a manner similar to that used in Theorem
14 to show that condition ( 3 ) of that theorem implies ( 4 ) . From Corollary 19 it
follows that S may be embedded as a dense subset of a compact Hausdorff
space ...
That ( 1 ) implies ( 2 ) can be proved in a manner similar to that used in Theorem
14 to show that condition ( 3 ) of that theorem implies ( 4 ) . From Corollary 19 it
follows that S may be embedded as a dense subset of a compact Hausdorff
space ...
Page 430
It is readily seen that ( iii ) implies ( ii ) . The other implications we desire are
decidedly non - trivial ; we will complete the proof by showing first that ( ii )
implies ( i ) , and then that ( i ) implies ( iii ) . Proof that condition ( ii ) implies ( i ) .
Let { Xn } ...
It is readily seen that ( iii ) implies ( ii ) . The other implications we desire are
decidedly non - trivial ; we will complete the proof by showing first that ( ii )
implies ( i ) , and then that ( i ) implies ( iii ) . Proof that condition ( ii ) implies ( i ) .
Let { Xn } ...
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Contents
Preliminary Concepts A Settheoretic Preliminaries 1 Notation and Elementary Notions | 1 |
Partially Ordered Systems | 7 |
Exercises | 9 |
Copyright | |
35 other sections not shown
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Linear Operators, Part 1 Nelson Dunford,Jacob T. Schwartz,William G. Bade,Robert G. Bartle Snippet view - 1958 |
Common terms and phrases
algebra analytic applied arbitrary assumed B-space ba(S Borel bounded called Chapter clear closed compact complex condition Consequently constant contains continuous functions converges Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hausdorff Hence Hilbert space identity implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear space mapping Math means measure space neighborhood norm obtained operator positive measure preceding projection PROOF properties proved range reflexive regular respect satisfies scalar seen separable sequence sequentially set function Show shown statement subset subspace sufficient Suppose Theorem theory tion topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero
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 |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of Michigan |
| Digitized | 20 Nov 2007 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |