Linear Operators: General theory |
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Page 147
... vanishes almost everywhere . If ƒ is μ - integrable then f ( s ) μ ( ds ) = 0 for every E in Σ if and only if f f vanishes almost everywhere . E PROOF . It is clear that a function vanishing almost everywhere is a null function ...
... vanishes almost everywhere . If ƒ is μ - integrable then f ( s ) μ ( ds ) = 0 for every E in Σ if and only if f f vanishes almost everywhere . E PROOF . It is clear that a function vanishing almost everywhere is a null function ...
Page 572
... vanishes identically on S ( α , ɛ ( α ; ) ) . Thus , if U is the union of those spheres S ( α1 , ε ( α ; ) ) which contain an infinite number of points of o ( T ) , then f vanishes identi- cally on U1 . Hence , U1 contains all but a ...
... vanishes identically on S ( α , ɛ ( α ; ) ) . Thus , if U is the union of those spheres S ( α1 , ε ( α ; ) ) which contain an infinite number of points of o ( T ) , then f vanishes identi- cally on U1 . Hence , U1 contains all but a ...
Page 593
... vanishes at certain points of o ( T ) . The next theorem is concerned with this possibility . 1 THEOREM . Let f , fn be in F ( T ) , and let { f ( T ) fn ( T ) } converge to zero in the uniform operator topology . Let f vanish at a ...
... vanishes at certain points of o ( T ) . The next theorem is concerned with this possibility . 1 THEOREM . Let f , fn be in F ( T ) , and let { f ( T ) fn ( T ) } converge to zero in the uniform operator topology . Let f vanish at a ...
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