Linear Operators: General theory |
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Page 42
... proves the statement made above . We have seen that the mapping a → H ( x ) is a homomorphism . To show that it is an isomorphism , let x 。 yo . We will prove that there exists an h 。€ H with h 。( x 。) ‡ ho ( yo ) . If xoyo , then ...
... proves the statement made above . We have seen that the mapping a → H ( x ) is a homomorphism . To show that it is an isomorphism , let x 。 yo . We will prove that there exists an h 。€ H with h 。( x 。) ‡ ho ( yo ) . If xoyo , then ...
Page 154
... prove a useful reciprocal relation between Stieltjes integrals with respect to two different functions of bounded variation , which is given by the famous formula for integration by parts . Before stating it , we prove a preliminary ...
... prove a useful reciprocal relation between Stieltjes integrals with respect to two different functions of bounded variation , which is given by the famous formula for integration by parts . Before stating it , we prove a preliminary ...
Page 495
... prove that Bo is an algebra under the natural product ( fg ) ( s ) = f ( s ) g ( s ) , se S. To see this , let h be a fixed element of C ( S ) and let B ( h ) { fe Both € Bo } . Evidently B ( h ) Bo , and it is clear that B ( h ) ...
... prove that Bo is an algebra under the natural product ( fg ) ( s ) = f ( s ) g ( s ) , se S. To see this , let h be a fixed element of C ( S ) and let B ( h ) { fe Both € Bo } . Evidently B ( h ) Bo , and it is clear that B ( h ) ...
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