Linear Operators: General theory |
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Page 410
... convex if x , y € K , and 0 ≤ a ≤ 1 , imply ax + ( 1 - a ) y € 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 . As ...
... convex if x , y € K , and 0 ≤ a ≤ 1 , imply ax + ( 1 - a ) y € 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 . As ...
Page 461
... convex set K which is compact in the X * topology of the normed linear space X , can be separated from an arbitrary closed convex set disjoint from K ( com- pare Theorem 2.10 ) . Tukey observed that this result implies that in a ...
... convex set K which is compact in the X * topology of the normed linear space X , can be separated from an arbitrary closed convex set disjoint from K ( com- pare Theorem 2.10 ) . Tukey observed that this result implies that in a ...
Page 842
... Convex combination , V.2.2 ( 414 ) . See also Convex hull , Convex set , Convex space ) Convex function , definition , VI.10.1 ( 520 ) study of , VI.10 Convex hull , V.2.2 ( 414 ) Convex set , II.4.1 ( 70 ) definition , V.1.1 ( 410 ) ...
... Convex combination , V.2.2 ( 414 ) . See also Convex hull , Convex set , Convex space ) Convex function , definition , VI.10.1 ( 520 ) study of , VI.10 Convex hull , V.2.2 ( 414 ) Convex set , II.4.1 ( 70 ) definition , V.1.1 ( 410 ) ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ