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 483
If T is weakly compact , then TS is compact in the Y * topology of Y and thus x ( TS
) is compact and hence closed in the Y * topology of Y * * . Thus if T is weakly
compact , ( i ) yields T * * ( S ) C x ( TS ) . According to Theorem V . 4 . 5 , S , = $ *
* ...
If T is weakly compact , then TS is compact in the Y * topology of Y and thus x ( TS
) is compact and hence closed in the Y * topology of Y * * . Thus if T is weakly
compact , ( i ) yields T * * ( S ) C x ( TS ) . According to Theorem V . 4 . 5 , S , = $ *
* ...
Page 494
From IV.10.2 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. given ...
From IV.10.2 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. given ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
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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 |