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 u , it follows that for such a function g we have √ 2 8 ( 82 ) μ2 ( d $ 2 ) = √ 2 - 1 ( 1 ) ...
... follows immediately . Clearly , g ( p ( ) ) is a u - 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 √ 2 8 ( 82 ) μ2 ( d $ 2 ) = √ 2 - 1 ( 1 ) ...
Page 403
... follows easily from ( 1 ) that ( 2 ) μ11 ( еÑ1 ) = μs ( 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 ...
... follows easily from ( 1 ) that ( 2 ) μ11 ( еÑ1 ) = μs ( 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 ...
Page 714
... follows from its definition that Ep is a projection , that EPED EDEP 0 , and that I Ep + ED . Further T commutes ... follows that Txp 0 and Taxp → 0 ; but since { T } has infinitely many terms equal to xp , it follows that ap = 0 ...
... follows from its definition that Ep is a projection , that EPED EDEP 0 , and that I Ep + ED . Further T commutes ... follows that Txp 0 and Taxp → 0 ; but since { T } has infinitely many terms equal to xp , it follows that ap = 0 ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ