Linear Operators: General theory |
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Page 15
... disjoint neighborhoods of x and y . ( c ) For every closed set A , and every x & A , there are disjoint neighborhoods of A and x . ( d ) For every pair of disjoint closed sets A and B , there are dis- joint neighborhoods of A and B. 2 ...
... disjoint neighborhoods of x and y . ( c ) For every closed set A , and every x & A , there are disjoint neighborhoods of A and x . ( d ) For every pair of disjoint closed sets A and B , there are dis- joint neighborhoods of A and B. 2 ...
Page 320
... disjoint sets in Σ . A number of elementary properties of the semi - variation are listed in the next lemma . 4 LEMMA . Let μ be a vector valued measure . Then ( a ) || μ || ( E ) ≥ ¦μ ( E ) | ≥ 0 , ΕΕΣ ; E e 2 ; ( b ) || u | ( E ) ...
... disjoint sets in Σ . A number of elementary properties of the semi - variation are listed in the next lemma . 4 LEMMA . Let μ be a vector valued measure . Then ( a ) || μ || ( E ) ≥ ¦μ ( E ) | ≥ 0 , ΕΕΣ ; E e 2 ; ( b ) || u | ( E ) ...
Page 461
... disjoint from K ( com- pare Theorem 2.10 ) . Tukey observed that this result implies that in a reflexive B - space two disjoint closed convex sets , one of which is bounded , can be separated by a hyperplane . Tukey demonstrated that ...
... disjoint from K ( com- pare Theorem 2.10 ) . Tukey observed that this result implies that in a reflexive B - space two disjoint closed convex sets , one of which is bounded , can be separated by a hyperplane . Tukey demonstrated that ...
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