Linear Operators: General theory |
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Page 10
... containing p . A neighborhood of the set A is an open set containing A. If A is a subset of X , then a point p is a limit point , or a point of accumulation , of A provided every neighborhood of p contains at least one point qp , with ...
... containing p . A neighborhood of the set A is an open set containing A. If A is a subset of X , then a point p is a limit point , or a point of accumulation , of A provided every neighborhood of p contains at least one point qp , with ...
Page 20
... contains a non - void open set . Thus A1 # X , and A1 is open , and contains a sphere S1 S ( P1 , 1 ) with 0 < & < 1/2 . By assumption , the set A2 does not contain the open set S ( p1 , ε1 / 2 ) ; hence the non - void open set AS ( P1 ...
... contains a non - void open set . Thus A1 # X , and A1 is open , and contains a sphere S1 S ( P1 , 1 ) with 0 < & < 1/2 . By assumption , the set A2 does not contain the open set S ( p1 , ε1 / 2 ) ; hence the non - void open set AS ( P1 ...
Page 309
... contains a set F with 0 < μ ( F ) ≤ ɛ . Sup- pose , on the contrary , that some such set B contains no set of Σ with positive measure at most ɛ . Then B is not an atom and thus contains a set G1 in with 0 < μ ( G1 ) < μ ( B ) . The set ...
... contains a set F with 0 < μ ( F ) ≤ ɛ . Sup- pose , on the contrary , that some such set B contains no set of Σ with positive measure at most ɛ . Then B is not an atom and thus contains a set G1 in with 0 < μ ( G1 ) < μ ( B ) . The set ...
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 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ