Linear Operators: General theory |
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Page 147
... vanishes almost everywhere . If f is u - integrable then † ( s ) μ ( ds ) = 0 for every E in Σ if and only if f vanishes almost everywhere . En PROOF . It is clear that a function vanishing almost everywhere is a null function ...
... vanishes almost everywhere . If f is u - integrable then † ( s ) μ ( ds ) = 0 for every E in Σ if and only if f vanishes almost everywhere . En PROOF . It is clear that a function vanishing almost everywhere is a null function ...
Page 572
... vanishes identically on S ( α , ε ( α ; ) ) . Thus , if U1 is the union of those spheres S ( a1 , ε ( α ) ) which contain an infinite number of points of o ( T ) , then ƒ vanishes identi- cally on U1 . Hence , U1 contains all but a ...
... vanishes identically on S ( α , ε ( α ; ) ) . Thus , if U1 is the union of those spheres S ( a1 , ε ( α ) ) which contain an infinite number of points of o ( T ) , then ƒ vanishes identi- cally on U1 . Hence , U1 contains all but a ...
Page 689
... vanishing on the ranges of all the operators T ( u ) —I then x * x * A ( u ) = x * E and thus a * vanishes on the range of E ' . The desired conclusion follows from Corollary II.3.13 . Q.E.D. = x * T ( u ) = 3 COROLLARY . In a reflexive ...
... vanishing on the ranges of all the operators T ( u ) —I then x * x * A ( u ) = x * E and thus a * vanishes on the range of E ' . The desired conclusion follows from Corollary II.3.13 . Q.E.D. = x * T ( u ) = 3 COROLLARY . In a reflexive ...
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