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 , E e 2 ; ( b ) || u || ( E ) ≤ 4 ...
... 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 , E e 2 ; ( b ) || u || ( E ) ≤ 4 ...
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
Preliminary Concepts | 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 ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ