Linear Operators: General theory |
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Page 311
... isomorphism of B ( S , Σ ) with C ( S1 ) where S1 is a particular compact Hausdorff space . We shall now investigate what properties the existence of this isomorphism implies for S1 and investigate in more detail the isomorphism of ba ...
... isomorphism of B ( S , Σ ) with C ( S1 ) where S1 is a particular compact Hausdorff space . We shall now investigate what properties the existence of this isomorphism implies for S1 and investigate in more detail the isomorphism of ba ...
Page 312
... isomorphism τ of the field Σ onto the field of all open and closed sets in S1 , i.e. , T ( E UF ) = t ( E ) Ut ( F ) , t ( EF ) = t ( E ) t ( F ) , and τ ( E ' ) = τ ( E ) ' for all E , F € Σ . α , PROOF . As before , let H be the ...
... isomorphism τ of the field Σ onto the field of all open and closed sets in S1 , i.e. , T ( E UF ) = t ( E ) Ut ( F ) , t ( EF ) = t ( E ) t ( F ) , and τ ( E ' ) = τ ( E ) ' for all E , F € Σ . α , PROOF . As before , let H be the ...
Page 313
... isomorphism of ba ( S1 , E1 ) onto ca ( S1 , E2 ) . ( c ) If E1 is in Σ1 then v ( μ1 , E1 ) = v ( U ( μ1 ) , E1 ) for all Ալ in ba ( S1 , 21 ) . PROOF . Recalling that 7 is an isomorphism of Σ onto E1 , it is clear that the mapping T ...
... isomorphism of ba ( S1 , E1 ) onto ca ( S1 , E2 ) . ( c ) If E1 is in Σ1 then v ( μ1 , E1 ) = v ( U ( μ1 ) , E1 ) for all Ալ in ba ( S1 , 21 ) . PROOF . Recalling that 7 is an isomorphism of Σ onto E1 , it is clear that the mapping T ...
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