Linear Operators: General theory |
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Page 403
... follows easily from ( 1 ) that ( 2 ) 삧1 ( eÑ1 ) = 삧 ( e ) . Equation ( 2 ) enables us to define a Gauss measure in real Hilbert space as follows . Call a Borel subset e of H a cylinder set if there exists an orthogonal projection E ...
... follows easily from ( 1 ) that ( 2 ) 삧1 ( eÑ1 ) = 삧 ( e ) . Equation ( 2 ) enables us to define a Gauss measure in real Hilbert space as follows . Call a Borel subset e of H a cylinder set if there exists an orthogonal projection E ...
Page 576
... follows from Theorem 20 that σ = σ ( T ) = 4. Q.E.D. 22 THEOREM . Let 21 , . . . , λ be poles of T , let v1 , orders ... follows that g ( m ) ( 2 ; ) = f ( m ) ( 2 ; ) , for 1 ≤ i ≤ k and 0 ≤m < v1 . Thus the equation f ( T ) E ( 0 ) ...
... follows from Theorem 20 that σ = σ ( T ) = 4. Q.E.D. 22 THEOREM . Let 21 , . . . , λ be poles of T , let v1 , orders ... follows that g ( m ) ( 2 ; ) = f ( m ) ( 2 ; ) , for 1 ≤ i ≤ k and 0 ≤m < v1 . Thus the equation f ( T ) E ( 0 ) ...
Page 689
... follows ( IV.8.9 ) that the set { A ( x ) ƒ , 0 ≤ a } is weakly sequentially compact in L1 ( S , Σ , μ ) . The conclusion now follows from Theorem 1. Q.E.D. 100 = The remaining part of the section will be concerned exclusively with the ...
... follows ( IV.8.9 ) that the set { A ( x ) ƒ , 0 ≤ a } is weakly sequentially compact in L1 ( S , Σ , μ ) . The conclusion now follows from Theorem 1. Q.E.D. 100 = The remaining part of the section will be concerned exclusively with the ...
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