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 , 12 51 Ta ] = [ 01 | 7 ) > M [ 2 ] . 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 , 12 51 Ta ] = [ 01 | 7 ) > M [ 2 ] . 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 454
To each point pe C there corresponds a unique nearest point N ( P ) € K . To see
this , note that Lemma IV . 4 . 2 implies that if { ki } is a sequence in K such that
limina p - k ; [ = inf ker \ p - k | then { k ; } converges , say , to qeK . If { k { } is
another ...
To each point pe C there corresponds a unique nearest point N ( P ) € K . To see
this , note that Lemma IV . 4 . 2 implies that if { ki } is a sequence in K such that
limina p - k ; [ = inf ker \ p - k | then { k ; } converges , say , to qeK . If { k { } is
another ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
31 other sections not shown
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Common terms and phrases
algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory 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 7 of Pure and applied mathematics Interscience Press Pure and applied mathematics Wiley classics library |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of California |
| Digitized | 16 Mar 2011 |
| ISBN | 0470226056, 9780470226056 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |