Linear Operators: General theory |
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Page 183
... follows immediately . Clearly , g ( p ( ) ) is a μ - integrable simple function if g is a 2- integrable simple function . From the definition of u1 it follows that for such a function g we have √ £ 8 ( 82 ) μ2 ( d $ 2 ) = √2-1 ( £ ) 8 ...
... follows immediately . Clearly , g ( p ( ) ) is a μ - integrable simple function if g is a 2- integrable simple function . From the definition of u1 it follows that for such a function g we have √ £ 8 ( 82 ) μ2 ( d $ 2 ) = √2-1 ( £ ) 8 ...
Page 576
... follows from Theorem 20 that σ = σ ( T . ) o ( T ) = 4. Q.E.D. ... 9 V1 , ... , Vk be their 22 THEOREM . Let λ1 ,. λ ... 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 . ) o ( T ) = 4. Q.E.D. ... 9 V1 , ... , Vk be their 22 THEOREM . Let λ1 ,. λ ... 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 { 4 ( a ) f , 0 ≤ a } is weakly sequentially compact in L1 ( S , Σ , μ ) . The conclusion now follows from Theorem 1. Q.E.D. The remaining part of the section will be concerned exclusively with the case ...
... follows ( IV.8.9 ) that the set { 4 ( a ) f , 0 ≤ a } is weakly sequentially compact in L1 ( S , Σ , μ ) . The conclusion now follows from Theorem 1. Q.E.D. The remaining part of the section will be concerned exclusively with the case ...
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 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ