Linear Operators: General theory |
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Page 17
A topological space X is said to be locally compact if every point has a
neighborhood whose closure is compact . A family of sets has the finite
intersection property if every finite subfamily has a non - void intersection . A
subset of X is called ...
A topological space X is said to be locally compact if every point has a
neighborhood whose closure is compact . A family of sets has the finite
intersection property if every finite subfamily has a non - void intersection . A
subset of X is called ...
Page 485
( Gantmacher ) An operator in B ( X , Y ) is weakly compact if and only if its adjoint
is weakly compact . PROOF . Let T be weakly compact . Since the closed unit
sphere S * of Y * is Y - compact ( V . 4 . 2 ) , it follows from Lemma 7 and Lemma 1
.
( Gantmacher ) An operator in B ( X , Y ) is weakly compact if and only if its adjoint
is weakly compact . PROOF . Let T be weakly compact . Since the closed unit
sphere S * of Y * is Y - compact ( V . 4 . 2 ) , it follows from Lemma 7 and Lemma 1
.
Page 494
2 we conclude that T * maps the unit sphere of X * into a conditionally weakly
compact set of rca ( S ) , and therefore T * is a weakly compact operator . By
Theorem 4 . 8 this implies that T is a weakly compact operator . Q . E . D . 4
THEOREM .
2 we conclude that T * maps the unit sphere of X * into a conditionally weakly
compact set of rca ( S ) , and therefore T * is a weakly compact operator . By
Theorem 4 . 8 this implies that T is a weakly compact operator . Q . E . D . 4
THEOREM .
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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 |