Linear Operators: General theory |
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Page 183
... follows immediately . Clearly , g ( p ( ) ) is a u - integrable simple function if g is a 2- integrable simple function . From the definition of μ , it follows that for such a function g we have √ £ 8 ( §2 ) μ2 ( d $ 2 ) = √ ̧ - 12 ...
... follows immediately . Clearly , g ( p ( ) ) is a u - integrable simple function if g is a 2- integrable simple function . From the definition of μ , it follows that for such a function g we have √ £ 8 ( §2 ) μ2 ( d $ 2 ) = √ ̧ - 12 ...
Page 403
... follows easily from ( 1 ) that ( 2 ) μÑ1 ( е§1 ) = μÑ 。( 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 ) μÑ1 ( е§1 ) = μÑ 。( 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 that g ( m ) ( λ ; ) = f ( m ) ( λ ) , for 1 ≤ i ≤k and 0 ≤m < v ;. Thus the equation f ( T ) E ( o ) = g ( T ) E ( 6 ) follows from Theorem 16. Q.E.D. 23 COROLLARY . Let the functions f , fn , n = 1 , 2 , . . . , be in F ( T ) ...
... follows that g ( m ) ( λ ; ) = f ( m ) ( λ ) , for 1 ≤ i ≤k and 0 ≤m < v ;. Thus the equation f ( T ) E ( o ) = g ( T ) E ( 6 ) follows from Theorem 16. Q.E.D. 23 COROLLARY . Let the functions f , fn , n = 1 , 2 , . . . , be in F ( T ) ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ