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 ; ( b ) || μ || ( E ) ≤ 4 sup ...
... 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 ; ( b ) || μ || ( E ) ≤ 4 sup ...
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
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ