Linear Operators: General theory |
From inside the book
Results 1-3 of 79
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 ...
What people are saying - Write a review
We haven't found any reviews in the usual places.
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
Copyright | |
31 other sections not shown
Other editions - View all
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