Linear Operators: General theory |
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Page 147
... vanishes almost everywhere . If f is u - integrable then [ f ( s ) u ( ds ) = 0 for every E in Σ if and only if f vanishes almost everywhere . the set En = PROOF . It is clear that a function vanishing almost everywhere is a null ...
... vanishes almost everywhere . If f is u - integrable then [ f ( s ) u ( ds ) = 0 for every E in Σ if and only if f vanishes almost everywhere . the set En = PROOF . It is clear that a function vanishing almost everywhere is a null ...
Page 302
... vanishes μ - almost every- where outside of a set of o - finite measure . We may therefore suppose that ( S , E , μ ) ... vanish outside S and obtain an h2 e L ( S , 2 , μ ) . Now for u - almost all s e S , 1 n = the sequence { h , ( s ) ...
... vanishes μ - almost every- where outside of a set of o - finite measure . We may therefore suppose that ( S , E , μ ) ... vanish outside S and obtain an h2 e L ( S , 2 , μ ) . Now for u - almost all s e S , 1 n = the sequence { h , ( s ) ...
Page 572
... vanishes identically on S ( α ,, ɛ ( α ; ) ) . 1 = Thus , if U1 is the union of those spheres S ( α1 , ɛ ( α ; ) ) which contain an infinite number of points of o ( T ) , then ƒ vanishes identi- cally on U1 . Hence , U , contains all ...
... vanishes identically on S ( α ,, ɛ ( α ; ) ) . 1 = Thus , if U1 is the union of those spheres S ( α1 , ɛ ( α ; ) ) which contain an infinite number of points of o ( T ) , then ƒ vanishes identi- cally on U1 . Hence , U , contains all ...
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