Linear Operators: General theory |
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Page 410
1 Definition. 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.
1 Definition. 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 - 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 sa s 1 , 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
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
31 other sections not shown
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Common terms and phrases
algebra Amer analytic applied arbitrary assumed B-space Banach spaces bounded called clear closed compact operator complex condition Consequently constant contains continuous functions converges convex convex set Corollary countably additive defined DEFINITION denote dense determined differential dimensional disjoint domain element equation equivalent everywhere Exercise exists extension field finite follows formula function defined function f given Hence Hilbert space identity implies inequality integral interval Lebesgue Lemma limit linear functional linear operator linear space Math neighborhood norm operator operator topology problem projection PROOF properties proved range reflexive representation respect satisfies scalar seen semi-group separable sequence set function Show shown statement subset subspace sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit sphere valued vector weak weakly compact zero
Bibliographic information
| Title | Linear Operators: General theory Part 1 of Linear Operators, Jacob T. Schwartz Volume 7 of Pure and applied mathematics Interscience Press Pure and applied mathematics Wiley classics library |
| Authors | Nelson Dunford, Jacob T. Schwartz |
| 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 |