Linear Operators: General theory |
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Page 164
... restriction to E is in ca ( S , E ) . We define 22 ( E ) 2 ( E ) -2 ( E ) , E e 2 ; clearly λ2 ≥0 . If 22 is not purely finitely additive , there is a non - zero λ ' e ca ( S , E ) such that ' ≤λ - λ ; hence λ ≤ = Ε = + 2≤ and ...
... restriction to E is in ca ( S , E ) . We define 22 ( E ) 2 ( E ) -2 ( E ) , E e 2 ; clearly λ2 ≥0 . If 22 is not purely finitely additive , there is a non - zero λ ' e ca ( S , E ) such that ' ≤λ - λ ; hence λ ≤ = Ε = + 2≤ and ...
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 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 is non - negative ...
Page 168
... restriction μ1 of μ to Σ1 has the following properties : ( i ) the measure space ( S1 , E1 , M1 ) is o - finite ; ( ii ) the B - space L ( S1 , E1 , μ1 , X1 ) is separable ; ( iii ) GCL ( S1 , E1 , M1 , X1 ) . 1 , 2 , • • be PROOF . Let ...
... restriction μ1 of μ to Σ1 has the following properties : ( i ) the measure space ( S1 , E1 , M1 ) is o - finite ; ( ii ) the B - space L ( S1 , E1 , μ1 , X1 ) is separable ; ( iii ) GCL ( S1 , E1 , M1 , X1 ) . 1 , 2 , • • be PROOF . Let ...
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