Linear Operators: General theory |
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Page 164
... restriction to Σ is in ca ( S , 2 ) . We define 2 ( E ) 2 ( E ) —λ ( E ) , E € Σ ; clearly λ ≥ 0. If 2 is not purely finitely additive , there is a non - zero λ ' e ca ( S , E ) such that 2 ' ≤2-21 ; hence ≤ + ' and supμec μ ( S ) ...
... restriction to Σ is in ca ( S , 2 ) . We define 2 ( E ) 2 ( E ) —λ ( E ) , E € Σ ; clearly λ ≥ 0. If 2 is not purely finitely additive , there is a non - zero λ ' e ca ( S , E ) such that 2 ' ≤2-21 ; hence ≤ + ' and supμec μ ( S ) ...
Page 166
... restriction of μ to a subfield of Σ there is an- other type of restriction of common occurrence in integration theory . In the following discussion of this other type of restriction it is not necessary to assume that μ is non - negative ...
... restriction of μ to a subfield of Σ there is an- other type of restriction of common occurrence in integration theory . In the following discussion of this other type of restriction it is not necessary to assume that μ is non - negative ...
Page 660
... restriction is not really essential , and the reader will see readily that a slight rewording of the proofs given ( if , indeed , any is needed ) will establish the results of these sections for real B - spaces also . Only in Section 8 ...
... restriction is not really essential , and the reader will see readily that a slight rewording of the proofs given ( if , indeed , any is needed ) will establish the results of these sections for real B - spaces also . Only in Section 8 ...
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