Linear Operators, Part 1 |
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Page 54
A linear mapping of one F - space into another is continuous if and only if it maps
bounded sets into bounded sets . PROOF . Let X , Y be F - spaces , let T : X → Y
be linear and continuous , and let B C X be bounded . For every neighborhood V
...
A linear mapping of one F - space into another is continuous if and only if it maps
bounded sets into bounded sets . PROOF . Let X , Y be F - spaces , let T : X → Y
be linear and continuous , and let B C X be bounded . For every neighborhood V
...
Page 97
and to be bounded if u is bounded . The total variation v ( u ) of an additive set
function u is important because it dominates u in the sense that v ( u , E ) 2 ( u ( E
) for E € £ ; the reader should test his comprehension of Definition 4 below by ...
and to be bounded if u is bounded . The total variation v ( u ) of an additive set
function u is important because it dominates u in the sense that v ( u , E ) 2 ( u ( E
) for E € £ ; the reader should test his comprehension of Definition 4 below by ...
Page 345
if and only if it is bounded and thi ( s ) converges ( to fli ) ( s ) ) for each 8 in [ a , b ]
and each i = 0 , 1 , ... , p . ( c ) ( p is not weakly complete , and not reflexive . ( d ) A
subset A Ç CP is conditionally compact if and only if it is bounded and for each ...
if and only if it is bounded and thi ( s ) converges ( to fli ) ( s ) ) for each 8 in [ a , b ]
and each i = 0 , 1 , ... , p . ( c ) ( p is not weakly complete , and not reflexive . ( d ) A
subset A Ç CP is conditionally compact if and only if it is bounded and for each ...
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Contents
Preliminary Concepts | 1 |
The VitaliHahnSaks Theorem and Spaces of Measures | 7 |
B Topological Preliminaries | 10 |
Copyright | |
87 other sections not shown
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Common terms and phrases
Akad algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed complex Consequently contains converges convex Corollary defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space identity implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space Math measure space metric space neighborhood norm open set positive measure problem projection Proof properties proved range reflexive representation respect Russian satisfies scalar seen separable sequence set function Show shown sphere statement subset Suppose Theorem theory topological space topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero
Bibliographic information
| Title | Linear Operators, Part 1 Linear Operators, Jacob T. Schwartz Volume 7 of Mathematics, Pure, Applied Volume 7 of Pure and applied mathematics, ISSN 0079-8185 Volume 7 of Pure and applied mathematics. A series of texts and monographs |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| Publisher | Interscience Publishers, 1958 |
| Original from | the University of California |
| Digitized | 26 Jun 2018 |
| Length | 2592 pages |
| Export Citation | BiBTeX EndNote RefMan |