Linear Operators: General theory |
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Page 495
... satisfies properties ( i ) and ( ii ) above so that B ( h ) Bo . This proves that if he C ( S ) and fe Bo . then fhe Bo . Now let f be a fixed element of Bo , and let B ( f ) { g € Bofg € Bo } . We have just proved that C ( S ) B ( f ) ...
... satisfies properties ( i ) and ( ii ) above so that B ( h ) Bo . This proves that if he C ( S ) and fe Bo . then fhe Bo . Now let f be a fixed element of Bo , and let B ( f ) { g € Bofg € Bo } . We have just proved that C ( S ) B ( f ) ...
Page 498
... satisfies ( i ) then ( ii ) defines an operator Ton X to L1 ( S , Σ , μ ) whose norm satisfies ( iii ) . Furthermore T is weakly compact if and only if x * ( ) is countably additive on Σ in the strong topology of X * . PROOF . If , for ...
... satisfies ( i ) then ( ii ) defines an operator Ton X to L1 ( S , Σ , μ ) whose norm satisfies ( iii ) . Furthermore T is weakly compact if and only if x * ( ) is countably additive on Σ in the strong topology of X * . PROOF . If , for ...
Page 557
... satisfies R1 ( T ) 0. In the same way , the product R2 of all the factors ( 2-2 ) , where Bi min ( x ,, v ( 2 ; ) ) , satisfies R2 ( T ) = 0. Since any polynomial P having a zero of order v ( λ ) at each λ e σ ( T ) is divisible by R2 ...
... satisfies R1 ( T ) 0. In the same way , the product R2 of all the factors ( 2-2 ) , where Bi min ( x ,, v ( 2 ; ) ) , satisfies R2 ( T ) = 0. Since any polynomial P having a zero of order v ( λ ) at each λ e σ ( T ) is divisible by R2 ...
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