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Page 410
A set K Q X is convex ifx,ye K, and 0 ^ a ^ 1, imply ax-\-(l — a)y e K. The following lemma is an obvious consequence of Definition 1. 2 Lemma. The intersection of an arbitrary family of convex subsets of the linear space X is convex.
A set K Q X is convex ifx,ye K, and 0 ^ a ^ 1, imply ax-\-(l — a)y e K. The following lemma is an obvious consequence of Definition 1. 2 Lemma. The intersection of an arbitrary family of convex subsets of the linear space X is convex.
Page 410
A set K CX is convex if x , y eK , and 0 sa s 1 , imply ax + ( 1 - aly e K. The following lemma is an obvious consequence of Definition 1 . 2 LEMMA . The intersection of an arbitrary family of convex subsets of the linear space X is ...
A set K CX is convex if x , y eK , and 0 sa s 1 , imply ax + ( 1 - aly e K. The following lemma is an obvious consequence of Definition 1 . 2 LEMMA . The intersection of an arbitrary family of convex subsets of the linear space X is ...
Page 461
and an arbitrary convex set is possible , provided they are disjoint ( compare Theorem 2.8 ) . He also proved that a convex set K which is compact in the X * topology of the normed linear space X , can be separated from an arbitrary ...
and an arbitrary convex set is possible , provided they are disjoint ( compare Theorem 2.8 ) . He also proved that a convex set K which is compact in the X * topology of the normed linear space X , can be separated from an arbitrary ...
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
quences | 26 |
Copyright | |
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Akad algebra Amer analytic applied arbitrary assume B-space Banach Banach spaces bounded called clear closed compact complex Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence implies inequality integral interval isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math mean measure space metric space neighborhood norm o-field open set operator positive problem Proc PROOF properties proved range regular respect Russian satisfies scalar seen 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
Bibliographic information
| Title | Linear Operators: General theory Part 1 of Linear Operators, Jacob T. Schwartz Volume 1 of Linear Operators: General Theory. Spectral Theory, Self Adjoint Operators in Hilbert Space. Spectral Operators. I. II. III, William George Bade Volume 7 of Pure & Applied Mathematics : a Wiley-interscience series of texts, monographs & tracts 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, Robert G. Bartle |
| Editors | William G. Bade, Robert G. Bartle |
| Contributor | Karreman Mathematics Research Collection |
| Edition | reprint |
| 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 |