Linear Operators: General theory |
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Page 183
... follows immediately . Clearly , g ( $ ( · ) ) is a μ1 - integrable simple function if g is a 2- integrable simple function . From the definition of u , it follows that for such a function g we have [ £ 8 ( 82 ) μ2 ( d $ q ) = √4-1 ( g ) ...
... follows immediately . Clearly , g ( $ ( · ) ) is a μ1 - integrable simple function if g is a 2- integrable simple function . From the definition of u , it follows that for such a function g we have [ £ 8 ( 82 ) μ2 ( d $ q ) = √4-1 ( g ) ...
Page 403
... follows easily from ( 1 ) that ( 2 ) = Mo ( e ) . Equation ( 2 ) enables us to define a Gauss measure in real Hilbert space as follows . Call a Borel subset e of a cylinder set if there exists an orthogonal projection E of 5 onto a ...
... follows easily from ( 1 ) that ( 2 ) = Mo ( e ) . Equation ( 2 ) enables us to define a Gauss measure in real Hilbert space as follows . Call a Borel subset e of a cylinder set if there exists an orthogonal projection E of 5 onto a ...
Page 714
... follows from its definition that Ep is a projection , that EPED 0 , and that I = Ep + Ep . Further T commutes with ... follows that T " xp → 0 and Tap → 0 ; but since XD Tæ { T } has infinitely many terms equal to ap , it follows that ...
... follows from its definition that Ep is a projection , that EPED 0 , and that I = Ep + Ep . Further T commutes with ... follows that T " xp → 0 and Tap → 0 ; but since XD Tæ { T } has infinitely many terms equal to ap , it follows that ...
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian 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