Linear Operators: General theory |
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Page 144
... bounded variation . Suppose that ƒ is finite valued , real , monotone increasing , and continuous on the right in ( —∞ , + ∞ ) . By our previous construc- tion , we have already assigned a non - negative measure u to every bounded Borel ...
... bounded variation . Suppose that ƒ is finite valued , real , monotone increasing , and continuous on the right in ( —∞ , + ∞ ) . By our previous construc- tion , we have already assigned a non - negative measure u to every bounded Borel ...
Page 381
... bounded ; ( 3 ) every continuous real valued function on S assumes its maximum ; ( 4 ) every bounded equicontinuous ... Borel functions . Once this extension is achieved the measure is readily obtained , and the func- tional represented ...
... bounded ; ( 3 ) every continuous real valued function on S assumes its maximum ; ( 4 ) every bounded equicontinuous ... Borel functions . Once this extension is achieved the measure is readily obtained , and the func- tional represented ...
Page 495
... Borel sets in S. Then the space B ( S , B ) ( cf. IV.2.12 ) is the space of bounded Borel measurable functions on S. We let Bo be the intersection of all linear manifolds BC B ( S , B ) with the two properties : ( i ) C ( S ) CB , ( ii ) ...
... Borel sets in S. Then the space B ( S , B ) ( cf. IV.2.12 ) is the space of bounded Borel measurable functions on S. We let Bo be the intersection of all linear manifolds BC B ( S , B ) with the two properties : ( i ) C ( S ) CB , ( ii ) ...
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