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Page 410
A set K Q X is convex iix,ye K, and 0 52 a 52 1, imply ax-\-(\—a)y eK. 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 iix,ye K, and 0 52 a 52 1, imply ax-\-(\—a)y eK. 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 Sasi , imply ax + ( 1 - 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 CX is convex if x , y eK , and 0 Sasi , imply ax + ( 1 - 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 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
closed ...
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
closed ...
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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 |